MLPY/Lib/site-packages/sympy/functions/elementary/trigonometric.py

from typing import Tuple as tTuple, Union as tUnion
from sympy.core.add import Add
from sympy.core.cache import cacheit
from sympy.core.expr import Expr
from sympy.core.function import Function, ArgumentIndexError, PoleError, expand_mul
from sympy.core.logic import fuzzy_not, fuzzy_or, FuzzyBool, fuzzy_and
from sympy.core.mod import Mod
from sympy.core.numbers import Rational, pi, Integer, Float, equal_valued
from sympy.core.relational import Ne, Eq
from sympy.core.singleton import S
from sympy.core.symbol import Symbol, Dummy
from sympy.core.sympify import sympify
from sympy.functions.combinatorial.factorials import factorial, RisingFactorial
from sympy.functions.combinatorial.numbers import bernoulli, euler
from sympy.functions.elementary.complexes import arg as arg_f, im, re
from sympy.functions.elementary.exponential import log, exp
from sympy.functions.elementary.integers import floor
from sympy.functions.elementary.miscellaneous import sqrt, Min, Max
from sympy.functions.elementary.piecewise import Piecewise
from sympy.functions.elementary._trigonometric_special import (
    cos_table, ipartfrac, fermat_coords)
from sympy.logic.boolalg import And
from sympy.ntheory import factorint
from sympy.polys.specialpolys import symmetric_poly
from sympy.utilities.iterables import numbered_symbols


###############################################################################
########################## UTILITIES ##########################################
###############################################################################


def _imaginary_unit_as_coefficient(arg):
    """ Helper to extract symbolic coefficient for imaginary unit """
    if isinstance(arg, Float):
        return None
    else:
        return arg.as_coefficient(S.ImaginaryUnit)

###############################################################################
########################## TRIGONOMETRIC FUNCTIONS ############################
###############################################################################


class TrigonometricFunction(Function):
    """Base class for trigonometric functions. """

    unbranched = True
    _singularities = (S.ComplexInfinity,)

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational and fuzzy_not(s.args[0].is_zero):
                return False
        else:
            return s.is_rational

    def _eval_is_algebraic(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if fuzzy_not(self.args[0].is_zero) and self.args[0].is_algebraic:
                return False
            pi_coeff = _pi_coeff(self.args[0])
            if pi_coeff is not None and pi_coeff.is_rational:
                return True
        else:
            return s.is_algebraic

    def _eval_expand_complex(self, deep=True, **hints):
        re_part, im_part = self.as_real_imag(deep=deep, **hints)
        return re_part + im_part*S.ImaginaryUnit

    def _as_real_imag(self, deep=True, **hints):
        if self.args[0].is_extended_real:
            if deep:
                hints['complex'] = False
                return (self.args[0].expand(deep, **hints), S.Zero)
            else:
                return (self.args[0], S.Zero)
        if deep:
            re, im = self.args[0].expand(deep, **hints).as_real_imag()
        else:
            re, im = self.args[0].as_real_imag()
        return (re, im)

    def _period(self, general_period, symbol=None):
        f = expand_mul(self.args[0])
        if symbol is None:
            symbol = tuple(f.free_symbols)[0]

        if not f.has(symbol):
            return S.Zero

        if f == symbol:
            return general_period

        if symbol in f.free_symbols:
            if f.is_Mul:
                g, h = f.as_independent(symbol)
                if h == symbol:
                    return general_period/abs(g)

            if f.is_Add:
                a, h = f.as_independent(symbol)
                g, h = h.as_independent(symbol, as_Add=False)
                if h == symbol:
                    return general_period/abs(g)

        raise NotImplementedError("Use the periodicity function instead.")


@cacheit
def _table2():
    # If nested sqrt's are worse than un-evaluation
    # you can require q to be in (1, 2, 3, 4, 6, 12)
    # q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
    # expressions with 2 or fewer sqrt nestings.
    return {
        12: (3, 4),
        20: (4, 5),
        30: (5, 6),
        15: (6, 10),
        24: (6, 8),
        40: (8, 10),
        60: (20, 30),
        120: (40, 60)
    }


def _peeloff_pi(arg):
    r"""
    Split ARG into two parts, a "rest" and a multiple of $\pi$.
    This assumes ARG to be an Add.
    The multiple of $\pi$ returned in the second position is always a Rational.

    Examples
    ========

    >>> from sympy.functions.elementary.trigonometric import _peeloff_pi
    >>> from sympy import pi
    >>> from sympy.abc import x, y
    >>> _peeloff_pi(x + pi/2)
    (x, 1/2)
    >>> _peeloff_pi(x + 2*pi/3 + pi*y)
    (x + pi*y + pi/6, 1/2)

    """
    pi_coeff = S.Zero
    rest_terms = []
    for a in Add.make_args(arg):
        K = a.coeff(pi)
        if K and K.is_rational:
            pi_coeff += K
        else:
            rest_terms.append(a)

    if pi_coeff is S.Zero:
        return arg, S.Zero

    m1 = (pi_coeff % S.Half)
    m2 = pi_coeff - m1
    if m2.is_integer or ((2*m2).is_integer and m2.is_even is False):
        return Add(*(rest_terms + [m1*pi])), m2
    return arg, S.Zero


def _pi_coeff(arg: Expr, cycles: int = 1) -> tUnion[Expr, None]:
    r"""
    When arg is a Number times $\pi$ (e.g. $3\pi/2$) then return the Number
    normalized to be in the range $[0, 2]$, else `None`.

    When an even multiple of $\pi$ is encountered, if it is multiplying
    something with known parity then the multiple is returned as 0 otherwise
    as 2.

    Examples
    ========

    >>> from sympy.functions.elementary.trigonometric import _pi_coeff
    >>> from sympy import pi, Dummy
    >>> from sympy.abc import x
    >>> _pi_coeff(3*x*pi)
    3*x
    >>> _pi_coeff(11*pi/7)
    11/7
    >>> _pi_coeff(-11*pi/7)
    3/7
    >>> _pi_coeff(4*pi)
    0
    >>> _pi_coeff(5*pi)
    1
    >>> _pi_coeff(5.0*pi)
    1
    >>> _pi_coeff(5.5*pi)
    3/2
    >>> _pi_coeff(2 + pi)

    >>> _pi_coeff(2*Dummy(integer=True)*pi)
    2
    >>> _pi_coeff(2*Dummy(even=True)*pi)
    0

    """
    if arg is pi:
        return S.One
    elif not arg:
        return S.Zero
    elif arg.is_Mul:
        cx = arg.coeff(pi)
        if cx:
            c, x = cx.as_coeff_Mul()  # pi is not included as coeff
            if c.is_Float:
                # recast exact binary fractions to Rationals
                f = abs(c) % 1
                if f != 0:
                    p = -int(round(log(f, 2).evalf()))
                    m = 2**p
                    cm = c*m
                    i = int(cm)
                    if equal_valued(i, cm):
                        c = Rational(i, m)
                        cx = c*x
                else:
                    c = Rational(int(c))
                    cx = c*x
            if x.is_integer:
                c2 = c % 2
                if c2 == 1:
                    return x
                elif not c2:
                    if x.is_even is not None:  # known parity
                        return S.Zero
                    return Integer(2)
                else:
                    return c2*x
            return cx
    elif arg.is_zero:
        return S.Zero
    return None


class sin(TrigonometricFunction):
    r"""
    The sine function.

    Returns the sine of x (measured in radians).

    Explanation
    ===========

    This function will evaluate automatically in the
    case $x/\pi$ is some rational number [4]_.  For example,
    if $x$ is a multiple of $\pi$, $\pi/2$, $\pi/3$, $\pi/4$, and $\pi/6$.

    Examples
    ========

    >>> from sympy import sin, pi
    >>> from sympy.abc import x
    >>> sin(x**2).diff(x)
    2*x*cos(x**2)
    >>> sin(1).diff(x)
    0
    >>> sin(pi)
    0
    >>> sin(pi/2)
    1
    >>> sin(pi/6)
    1/2
    >>> sin(pi/12)
    -sqrt(2)/4 + sqrt(6)/4


    See Also
    ========

    csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Sin
    .. [4] https://mathworld.wolfram.com/TrigonometryAngles.html

    """

    def period(self, symbol=None):
        return self._period(2*pi, symbol)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return cos(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.sets.setexpr import SetExpr
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg.is_zero:
                return S.Zero
            elif arg in (S.Infinity, S.NegativeInfinity):
                return AccumBounds(-1, 1)

        if arg is S.ComplexInfinity:
            return S.NaN

        if isinstance(arg, AccumBounds):
            from sympy.sets.sets import FiniteSet
            min, max = arg.min, arg.max
            d = floor(min/(2*pi))
            if min is not S.NegativeInfinity:
                min = min - d*2*pi
            if max is not S.Infinity:
                max = max - d*2*pi
            if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
                    is not S.EmptySet and \
                    AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2),
                        pi*Rational(7, 2))) is not S.EmptySet:
                return AccumBounds(-1, 1)
            elif AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(5, 2))) \
                    is not S.EmptySet:
                return AccumBounds(Min(sin(min), sin(max)), 1)
            elif AccumBounds(min, max).intersection(FiniteSet(pi*Rational(3, 2), pi*Rational(8, 2))) \
                        is not S.EmptySet:
                return AccumBounds(-1, Max(sin(min), sin(max)))
            else:
                return AccumBounds(Min(sin(min), sin(max)),
                                Max(sin(min), sin(max)))
        elif isinstance(arg, SetExpr):
            return arg._eval_func(cls)

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import sinh
            return S.ImaginaryUnit*sinh(i_coeff)

        pi_coeff = _pi_coeff(arg)
        if pi_coeff is not None:
            if pi_coeff.is_integer:
                return S.Zero

            if (2*pi_coeff).is_integer:
                # is_even-case handled above as then pi_coeff.is_integer,
                # so check if known to be not even
                if pi_coeff.is_even is False:
                    return S.NegativeOne**(pi_coeff - S.Half)

            if not pi_coeff.is_Rational:
                narg = pi_coeff*pi
                if narg != arg:
                    return cls(narg)
                return None

            # https://github.com/sympy/sympy/issues/6048
            # transform a sine to a cosine, to avoid redundant code
            if pi_coeff.is_Rational:
                x = pi_coeff % 2
                if x > 1:
                    return -cls((x % 1)*pi)
                if 2*x > 1:
                    return cls((1 - x)*pi)
                narg = ((pi_coeff + Rational(3, 2)) % 2)*pi
                result = cos(narg)
                if not isinstance(result, cos):
                    return result
                if pi_coeff*pi != arg:
                    return cls(pi_coeff*pi)
                return None

        if arg.is_Add:
            x, m = _peeloff_pi(arg)
            if m:
                m = m*pi
                return sin(m)*cos(x) + cos(m)*sin(x)

        if arg.is_zero:
            return S.Zero

        if isinstance(arg, asin):
            return arg.args[0]

        if isinstance(arg, atan):
            x = arg.args[0]
            return x/sqrt(1 + x**2)

        if isinstance(arg, atan2):
            y, x = arg.args
            return y/sqrt(x**2 + y**2)

        if isinstance(arg, acos):
            x = arg.args[0]
            return sqrt(1 - x**2)

        if isinstance(arg, acot):
            x = arg.args[0]
            return 1/(sqrt(1 + 1/x**2)*x)

        if isinstance(arg, acsc):
            x = arg.args[0]
            return 1/x

        if isinstance(arg, asec):
            x = arg.args[0]
            return sqrt(1 - 1/x**2)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            if len(previous_terms) > 2:
                p = previous_terms[-2]
                return -p*x**2/(n*(n - 1))
            else:
                return S.NegativeOne**(n//2)*x**n/factorial(n)

    def _eval_nseries(self, x, n, logx, cdir=0):
        arg = self.args[0]
        if logx is not None:
            arg = arg.subs(log(x), logx)
        if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
            raise PoleError("Cannot expand %s around 0" % (self))
        return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        from sympy.functions.elementary.hyperbolic import HyperbolicFunction
        I = S.ImaginaryUnit
        if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
            arg = arg.func(arg.args[0]).rewrite(exp)
        return (exp(arg*I) - exp(-arg*I))/(2*I)

    def _eval_rewrite_as_Pow(self, arg, **kwargs):
        if isinstance(arg, log):
            I = S.ImaginaryUnit
            x = arg.args[0]
            return I*x**-I/2 - I*x**I /2

    def _eval_rewrite_as_cos(self, arg, **kwargs):
        return cos(arg - pi/2, evaluate=False)

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        tan_half = tan(S.Half*arg)
        return 2*tan_half/(1 + tan_half**2)

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return sin(arg)*cos(arg)/cos(arg)

    def _eval_rewrite_as_cot(self, arg, **kwargs):
        cot_half = cot(S.Half*arg)
        return Piecewise((0, And(Eq(im(arg), 0), Eq(Mod(arg, pi), 0))),
                         (2*cot_half/(1 + cot_half**2), True))

    def _eval_rewrite_as_pow(self, arg, **kwargs):
        return self.rewrite(cos, **kwargs).rewrite(pow, **kwargs)

    def _eval_rewrite_as_sqrt(self, arg, **kwargs):
        return self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs)

    def _eval_rewrite_as_csc(self, arg, **kwargs):
        return 1/csc(arg)

    def _eval_rewrite_as_sec(self, arg, **kwargs):
        return 1/sec(arg - pi/2, evaluate=False)

    def _eval_rewrite_as_sinc(self, arg, **kwargs):
        return arg*sinc(arg)

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return sqrt(pi*arg/2)*besselj(S.Half, arg)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        from sympy.functions.elementary.hyperbolic import cosh, sinh
        re, im = self._as_real_imag(deep=deep, **hints)
        return (sin(re)*cosh(im), cos(re)*sinh(im))

    def _eval_expand_trig(self, **hints):
        from sympy.functions.special.polynomials import chebyshevt, chebyshevu
        arg = self.args[0]
        x = None
        if arg.is_Add:  # TODO, implement more if deep stuff here
            # TODO: Do this more efficiently for more than two terms
            x, y = arg.as_two_terms()
            sx = sin(x, evaluate=False)._eval_expand_trig()
            sy = sin(y, evaluate=False)._eval_expand_trig()
            cx = cos(x, evaluate=False)._eval_expand_trig()
            cy = cos(y, evaluate=False)._eval_expand_trig()
            return sx*cy + sy*cx
        elif arg.is_Mul:
            n, x = arg.as_coeff_Mul(rational=True)
            if n.is_Integer:  # n will be positive because of .eval
                # canonicalization

                # See https://mathworld.wolfram.com/Multiple-AngleFormulas.html
                if n.is_odd:
                    return S.NegativeOne**((n - 1)/2)*chebyshevt(n, sin(x))
                else:
                    return expand_mul(S.NegativeOne**(n/2 - 1)*cos(x)*
                                      chebyshevu(n - 1, sin(x)), deep=False)
        return sin(arg)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = x0/pi
        if n.is_integer:
            lt = (arg - n*pi).as_leading_term(x)
            return (S.NegativeOne**n)*lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in [S.Infinity, S.NegativeInfinity]:
            return AccumBounds(-1, 1)
        return self.func(x0) if x0.is_finite else self

    def _eval_is_extended_real(self):
        if self.args[0].is_extended_real:
            return True

    def _eval_is_finite(self):
        arg = self.args[0]
        if arg.is_extended_real:
            return True

    def _eval_is_zero(self):
        rest, pi_mult = _peeloff_pi(self.args[0])
        if rest.is_zero:
            return pi_mult.is_integer

    def _eval_is_complex(self):
        if self.args[0].is_extended_real \
                or self.args[0].is_complex:
            return True


class cos(TrigonometricFunction):
    """
    The cosine function.

    Returns the cosine of x (measured in radians).

    Explanation
    ===========

    See :func:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import cos, pi
    >>> from sympy.abc import x
    >>> cos(x**2).diff(x)
    -2*x*sin(x**2)
    >>> cos(1).diff(x)
    0
    >>> cos(pi)
    -1
    >>> cos(pi/2)
    0
    >>> cos(2*pi/3)
    -1/2
    >>> cos(pi/12)
    sqrt(2)/4 + sqrt(6)/4

    See Also
    ========

    sin, csc, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Cos

    """

    def period(self, symbol=None):
        return self._period(2*pi, symbol)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -sin(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        from sympy.functions.special.polynomials import chebyshevt
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.sets.setexpr import SetExpr
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg.is_zero:
                return S.One
            elif arg in (S.Infinity, S.NegativeInfinity):
                # In this case it is better to return AccumBounds(-1, 1)
                # rather than returning S.NaN, since AccumBounds(-1, 1)
                # preserves the information that sin(oo) is between
                # -1 and 1, where S.NaN does not do that.
                return AccumBounds(-1, 1)

        if arg is S.ComplexInfinity:
            return S.NaN

        if isinstance(arg, AccumBounds):
            return sin(arg + pi/2)
        elif isinstance(arg, SetExpr):
            return arg._eval_func(cls)

        if arg.is_extended_real and arg.is_finite is False:
            return AccumBounds(-1, 1)

        if arg.could_extract_minus_sign():
            return cls(-arg)

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import cosh
            return cosh(i_coeff)

        pi_coeff = _pi_coeff(arg)
        if pi_coeff is not None:
            if pi_coeff.is_integer:
                return (S.NegativeOne)**pi_coeff

            if (2*pi_coeff).is_integer:
                # is_even-case handled above as then pi_coeff.is_integer,
                # so check if known to be not even
                if pi_coeff.is_even is False:
                    return S.Zero

            if not pi_coeff.is_Rational:
                narg = pi_coeff*pi
                if narg != arg:
                    return cls(narg)
                return None

            # cosine formula #####################
            # https://github.com/sympy/sympy/issues/6048
            # explicit calculations are performed for
            # cos(k pi/n) for n = 8,10,12,15,20,24,30,40,60,120
            # Some other exact values like cos(k pi/240) can be
            # calculated using a partial-fraction decomposition
            # by calling cos( X ).rewrite(sqrt)
            if pi_coeff.is_Rational:
                q = pi_coeff.q
                p = pi_coeff.p % (2*q)
                if p > q:
                    narg = (pi_coeff - 1)*pi
                    return -cls(narg)
                if 2*p > q:
                    narg = (1 - pi_coeff)*pi
                    return -cls(narg)

                # If nested sqrt's are worse than un-evaluation
                # you can require q to be in (1, 2, 3, 4, 6, 12)
                # q <= 12, q=15, q=20, q=24, q=30, q=40, q=60, q=120 return
                # expressions with 2 or fewer sqrt nestings.
                table2 = _table2()
                if q in table2:
                    a, b = table2[q]
                    a, b = p*pi/a, p*pi/b
                    nvala, nvalb = cls(a), cls(b)
                    if None in (nvala, nvalb):
                        return None
                    return nvala*nvalb + cls(pi/2 - a)*cls(pi/2 - b)

                if q > 12:
                    return None

                cst_table_some = {
                    3: S.Half,
                    5: (sqrt(5) + 1) / 4,
                }
                if q in cst_table_some:
                    cts = cst_table_some[pi_coeff.q]
                    return chebyshevt(pi_coeff.p, cts).expand()

                if 0 == q % 2:
                    narg = (pi_coeff*2)*pi
                    nval = cls(narg)
                    if None == nval:
                        return None
                    x = (2*pi_coeff + 1)/2
                    sign_cos = (-1)**((-1 if x < 0 else 1)*int(abs(x)))
                    return sign_cos*sqrt( (1 + nval)/2 )
            return None

        if arg.is_Add:
            x, m = _peeloff_pi(arg)
            if m:
                m = m*pi
                return cos(m)*cos(x) - sin(m)*sin(x)

        if arg.is_zero:
            return S.One

        if isinstance(arg, acos):
            return arg.args[0]

        if isinstance(arg, atan):
            x = arg.args[0]
            return 1/sqrt(1 + x**2)

        if isinstance(arg, atan2):
            y, x = arg.args
            return x/sqrt(x**2 + y**2)

        if isinstance(arg, asin):
            x = arg.args[0]
            return sqrt(1 - x ** 2)

        if isinstance(arg, acot):
            x = arg.args[0]
            return 1/sqrt(1 + 1/x**2)

        if isinstance(arg, acsc):
            x = arg.args[0]
            return sqrt(1 - 1/x**2)

        if isinstance(arg, asec):
            x = arg.args[0]
            return 1/x

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)

            if len(previous_terms) > 2:
                p = previous_terms[-2]
                return -p*x**2/(n*(n - 1))
            else:
                return S.NegativeOne**(n//2)*x**n/factorial(n)

    def _eval_nseries(self, x, n, logx, cdir=0):
        arg = self.args[0]
        if logx is not None:
            arg = arg.subs(log(x), logx)
        if arg.subs(x, 0).has(S.NaN, S.ComplexInfinity):
            raise PoleError("Cannot expand %s around 0" % (self))
        return Function._eval_nseries(self, x, n=n, logx=logx, cdir=cdir)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        I = S.ImaginaryUnit
        from sympy.functions.elementary.hyperbolic import HyperbolicFunction
        if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
            arg = arg.func(arg.args[0]).rewrite(exp, **kwargs)
        return (exp(arg*I) + exp(-arg*I))/2

    def _eval_rewrite_as_Pow(self, arg, **kwargs):
        if isinstance(arg, log):
            I = S.ImaginaryUnit
            x = arg.args[0]
            return x**I/2 + x**-I/2

    def _eval_rewrite_as_sin(self, arg, **kwargs):
        return sin(arg + pi/2, evaluate=False)

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        tan_half = tan(S.Half*arg)**2
        return (1 - tan_half)/(1 + tan_half)

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return sin(arg)*cos(arg)/sin(arg)

    def _eval_rewrite_as_cot(self, arg, **kwargs):
        cot_half = cot(S.Half*arg)**2
        return Piecewise((1, And(Eq(im(arg), 0), Eq(Mod(arg, 2*pi), 0))),
                         ((cot_half - 1)/(cot_half + 1), True))

    def _eval_rewrite_as_pow(self, arg, **kwargs):
        return self._eval_rewrite_as_sqrt(arg, **kwargs)

    def _eval_rewrite_as_sqrt(self, arg: Expr, **kwargs):
        from sympy.functions.special.polynomials import chebyshevt

        pi_coeff = _pi_coeff(arg)
        if pi_coeff is None:
            return None

        if isinstance(pi_coeff, Integer):
            return None

        if not isinstance(pi_coeff, Rational):
            return None

        cst_table_some = cos_table()

        if pi_coeff.q in cst_table_some:
            rv = chebyshevt(pi_coeff.p, cst_table_some[pi_coeff.q]())
            if pi_coeff.q < 257:
                rv = rv.expand()
            return rv

        if not pi_coeff.q % 2:  # recursively remove factors of 2
            pico2 = pi_coeff * 2
            nval = cos(pico2 * pi).rewrite(sqrt, **kwargs)
            x = (pico2 + 1) / 2
            sign_cos = -1 if int(x) % 2 else 1
            return sign_cos * sqrt((1 + nval) / 2)

        FC = fermat_coords(pi_coeff.q)
        if FC:
            denoms = FC
        else:
            denoms = [b**e for b, e in factorint(pi_coeff.q).items()]

        apart = ipartfrac(*denoms)
        decomp = (pi_coeff.p * Rational(n, d) for n, d in zip(apart, denoms))
        X = [(x[1], x[0]*pi) for x in zip(decomp, numbered_symbols('z'))]
        pcls = cos(sum(x[0] for x in X))._eval_expand_trig().subs(X)

        if not FC or len(FC) == 1:
            return pcls
        return pcls.rewrite(sqrt, **kwargs)

    def _eval_rewrite_as_sec(self, arg, **kwargs):
        return 1/sec(arg)

    def _eval_rewrite_as_csc(self, arg, **kwargs):
        return 1/sec(arg).rewrite(csc, **kwargs)

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return Piecewise(
                (sqrt(pi*arg/2)*besselj(-S.Half, arg), Ne(arg, 0)),
                (1, True)
            )

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        from sympy.functions.elementary.hyperbolic import cosh, sinh
        re, im = self._as_real_imag(deep=deep, **hints)
        return (cos(re)*cosh(im), -sin(re)*sinh(im))

    def _eval_expand_trig(self, **hints):
        from sympy.functions.special.polynomials import chebyshevt
        arg = self.args[0]
        x = None
        if arg.is_Add:  # TODO: Do this more efficiently for more than two terms
            x, y = arg.as_two_terms()
            sx = sin(x, evaluate=False)._eval_expand_trig()
            sy = sin(y, evaluate=False)._eval_expand_trig()
            cx = cos(x, evaluate=False)._eval_expand_trig()
            cy = cos(y, evaluate=False)._eval_expand_trig()
            return cx*cy - sx*sy
        elif arg.is_Mul:
            coeff, terms = arg.as_coeff_Mul(rational=True)
            if coeff.is_Integer:
                return chebyshevt(coeff, cos(terms))
        return cos(arg)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = (x0 + pi/2)/pi
        if n.is_integer:
            lt = (arg - n*pi + pi/2).as_leading_term(x)
            return (S.NegativeOne**n)*lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in [S.Infinity, S.NegativeInfinity]:
            return AccumBounds(-1, 1)
        return self.func(x0) if x0.is_finite else self

    def _eval_is_extended_real(self):
        if self.args[0].is_extended_real:
            return True

    def _eval_is_finite(self):
        arg = self.args[0]

        if arg.is_extended_real:
            return True

    def _eval_is_complex(self):
        if self.args[0].is_extended_real \
            or self.args[0].is_complex:
            return True

    def _eval_is_zero(self):
        rest, pi_mult = _peeloff_pi(self.args[0])
        if rest.is_zero and pi_mult:
            return (pi_mult - S.Half).is_integer


class tan(TrigonometricFunction):
    """
    The tangent function.

    Returns the tangent of x (measured in radians).

    Explanation
    ===========

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import tan, pi
    >>> from sympy.abc import x
    >>> tan(x**2).diff(x)
    2*x*(tan(x**2)**2 + 1)
    >>> tan(1).diff(x)
    0
    >>> tan(pi/8).expand()
    -1 + sqrt(2)

    See Also
    ========

    sin, csc, cos, sec, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Tan

    """

    def period(self, symbol=None):
        return self._period(pi, symbol)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return S.One + self**2
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return atan

    @classmethod
    def eval(cls, arg):
        from sympy.calculus.accumulationbounds import AccumBounds
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg.is_zero:
                return S.Zero
            elif arg in (S.Infinity, S.NegativeInfinity):
                return AccumBounds(S.NegativeInfinity, S.Infinity)

        if arg is S.ComplexInfinity:
            return S.NaN

        if isinstance(arg, AccumBounds):
            min, max = arg.min, arg.max
            d = floor(min/pi)
            if min is not S.NegativeInfinity:
                min = min - d*pi
            if max is not S.Infinity:
                max = max - d*pi
            from sympy.sets.sets import FiniteSet
            if AccumBounds(min, max).intersection(FiniteSet(pi/2, pi*Rational(3, 2))):
                return AccumBounds(S.NegativeInfinity, S.Infinity)
            else:
                return AccumBounds(tan(min), tan(max))

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import tanh
            return S.ImaginaryUnit*tanh(i_coeff)

        pi_coeff = _pi_coeff(arg, 2)
        if pi_coeff is not None:
            if pi_coeff.is_integer:
                return S.Zero

            if not pi_coeff.is_Rational:
                narg = pi_coeff*pi
                if narg != arg:
                    return cls(narg)
                return None

            if pi_coeff.is_Rational:
                q = pi_coeff.q
                p = pi_coeff.p % q
                # ensure simplified results are returned for n*pi/5, n*pi/10
                table10 = {
                    1: sqrt(1 - 2*sqrt(5)/5),
                    2: sqrt(5 - 2*sqrt(5)),
                    3: sqrt(1 + 2*sqrt(5)/5),
                    4: sqrt(5 + 2*sqrt(5))
                    }
                if q in (5, 10):
                    n = 10*p/q
                    if n > 5:
                        n = 10 - n
                        return -table10[n]
                    else:
                        return table10[n]
                if not pi_coeff.q % 2:
                    narg = pi_coeff*pi*2
                    cresult, sresult = cos(narg), cos(narg - pi/2)
                    if not isinstance(cresult, cos) \
                            and not isinstance(sresult, cos):
                        if sresult == 0:
                            return S.ComplexInfinity
                        return 1/sresult - cresult/sresult

                table2 = _table2()
                if q in table2:
                    a, b = table2[q]
                    nvala, nvalb = cls(p*pi/a), cls(p*pi/b)
                    if None in (nvala, nvalb):
                        return None
                    return (nvala - nvalb)/(1 + nvala*nvalb)
                narg = ((pi_coeff + S.Half) % 1 - S.Half)*pi
                # see cos() to specify which expressions should  be
                # expanded automatically in terms of radicals
                cresult, sresult = cos(narg), cos(narg - pi/2)
                if not isinstance(cresult, cos) \
                        and not isinstance(sresult, cos):
                    if cresult == 0:
                        return S.ComplexInfinity
                    return (sresult/cresult)
                if narg != arg:
                    return cls(narg)

        if arg.is_Add:
            x, m = _peeloff_pi(arg)
            if m:
                tanm = tan(m*pi)
                if tanm is S.ComplexInfinity:
                    return -cot(x)
                else: # tanm == 0
                    return tan(x)

        if arg.is_zero:
            return S.Zero

        if isinstance(arg, atan):
            return arg.args[0]

        if isinstance(arg, atan2):
            y, x = arg.args
            return y/x

        if isinstance(arg, asin):
            x = arg.args[0]
            return x/sqrt(1 - x**2)

        if isinstance(arg, acos):
            x = arg.args[0]
            return sqrt(1 - x**2)/x

        if isinstance(arg, acot):
            x = arg.args[0]
            return 1/x

        if isinstance(arg, acsc):
            x = arg.args[0]
            return 1/(sqrt(1 - 1/x**2)*x)

        if isinstance(arg, asec):
            x = arg.args[0]
            return sqrt(1 - 1/x**2)*x

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            a, b = ((n - 1)//2), 2**(n + 1)

            B = bernoulli(n + 1)
            F = factorial(n + 1)

            return S.NegativeOne**a*b*(b - 1)*B/F*x**n

    def _eval_nseries(self, x, n, logx, cdir=0):
        i = self.args[0].limit(x, 0)*2/pi
        if i and i.is_Integer:
            return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
        return Function._eval_nseries(self, x, n=n, logx=logx)

    def _eval_rewrite_as_Pow(self, arg, **kwargs):
        if isinstance(arg, log):
            I = S.ImaginaryUnit
            x = arg.args[0]
            return I*(x**-I - x**I)/(x**-I + x**I)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        re, im = self._as_real_imag(deep=deep, **hints)
        if im:
            from sympy.functions.elementary.hyperbolic import cosh, sinh
            denom = cos(2*re) + cosh(2*im)
            return (sin(2*re)/denom, sinh(2*im)/denom)
        else:
            return (self.func(re), S.Zero)

    def _eval_expand_trig(self, **hints):
        arg = self.args[0]
        x = None
        if arg.is_Add:
            n = len(arg.args)
            TX = []
            for x in arg.args:
                tx = tan(x, evaluate=False)._eval_expand_trig()
                TX.append(tx)

            Yg = numbered_symbols('Y')
            Y = [ next(Yg) for i in range(n) ]

            p = [0, 0]
            for i in range(n + 1):
                p[1 - i % 2] += symmetric_poly(i, Y)*(-1)**((i % 4)//2)
            return (p[0]/p[1]).subs(list(zip(Y, TX)))

        elif arg.is_Mul:
            coeff, terms = arg.as_coeff_Mul(rational=True)
            if coeff.is_Integer and coeff > 1:
                I = S.ImaginaryUnit
                z = Symbol('dummy', real=True)
                P = ((1 + I*z)**coeff).expand()
                return (im(P)/re(P)).subs([(z, tan(terms))])
        return tan(arg)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        I = S.ImaginaryUnit
        from sympy.functions.elementary.hyperbolic import HyperbolicFunction
        if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
            arg = arg.func(arg.args[0]).rewrite(exp)
        neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
        return I*(neg_exp - pos_exp)/(neg_exp + pos_exp)

    def _eval_rewrite_as_sin(self, x, **kwargs):
        return 2*sin(x)**2/sin(2*x)

    def _eval_rewrite_as_cos(self, x, **kwargs):
        return cos(x - pi/2, evaluate=False)/cos(x)

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return sin(arg)/cos(arg)

    def _eval_rewrite_as_cot(self, arg, **kwargs):
        return 1/cot(arg)

    def _eval_rewrite_as_sec(self, arg, **kwargs):
        sin_in_sec_form = sin(arg).rewrite(sec, **kwargs)
        cos_in_sec_form = cos(arg).rewrite(sec, **kwargs)
        return sin_in_sec_form/cos_in_sec_form

    def _eval_rewrite_as_csc(self, arg, **kwargs):
        sin_in_csc_form = sin(arg).rewrite(csc, **kwargs)
        cos_in_csc_form = cos(arg).rewrite(csc, **kwargs)
        return sin_in_csc_form/cos_in_csc_form

    def _eval_rewrite_as_pow(self, arg, **kwargs):
        y = self.rewrite(cos, **kwargs).rewrite(pow, **kwargs)
        if y.has(cos):
            return None
        return y

    def _eval_rewrite_as_sqrt(self, arg, **kwargs):
        y = self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs)
        if y.has(cos):
            return None
        return y

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return besselj(S.Half, arg)/besselj(-S.Half, arg)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.functions.elementary.complexes import re
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = 2*x0/pi
        if n.is_integer:
            lt = (arg - n*pi/2).as_leading_term(x)
            return lt if n.is_even else -1/lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in (S.Infinity, S.NegativeInfinity):
            return AccumBounds(S.NegativeInfinity, S.Infinity)
        return self.func(x0) if x0.is_finite else self

    def _eval_is_extended_real(self):
        # FIXME: currently tan(pi/2) return zoo
        return self.args[0].is_extended_real

    def _eval_is_real(self):
        arg = self.args[0]
        if arg.is_real and (arg/pi - S.Half).is_integer is False:
            return True

    def _eval_is_finite(self):
        arg = self.args[0]

        if arg.is_real and (arg/pi - S.Half).is_integer is False:
            return True

        if arg.is_imaginary:
            return True

    def _eval_is_zero(self):
        rest, pi_mult = _peeloff_pi(self.args[0])
        if rest.is_zero:
            return pi_mult.is_integer

    def _eval_is_complex(self):
        arg = self.args[0]

        if arg.is_real and (arg/pi - S.Half).is_integer is False:
            return True


class cot(TrigonometricFunction):
    """
    The cotangent function.

    Returns the cotangent of x (measured in radians).

    Explanation
    ===========

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import cot, pi
    >>> from sympy.abc import x
    >>> cot(x**2).diff(x)
    2*x*(-cot(x**2)**2 - 1)
    >>> cot(1).diff(x)
    0
    >>> cot(pi/12)
    sqrt(3) + 2

    See Also
    ========

    sin, csc, cos, sec, tan
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Cot

    """

    def period(self, symbol=None):
        return self._period(pi, symbol)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return S.NegativeOne - self**2
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return acot

    @classmethod
    def eval(cls, arg):
        from sympy.calculus.accumulationbounds import AccumBounds
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            if arg.is_zero:
                return S.ComplexInfinity
            elif arg in (S.Infinity, S.NegativeInfinity):
                return AccumBounds(S.NegativeInfinity, S.Infinity)

        if arg is S.ComplexInfinity:
            return S.NaN

        if isinstance(arg, AccumBounds):
            return -tan(arg + pi/2)

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import coth
            return -S.ImaginaryUnit*coth(i_coeff)

        pi_coeff = _pi_coeff(arg, 2)
        if pi_coeff is not None:
            if pi_coeff.is_integer:
                return S.ComplexInfinity

            if not pi_coeff.is_Rational:
                narg = pi_coeff*pi
                if narg != arg:
                    return cls(narg)
                return None

            if pi_coeff.is_Rational:
                if pi_coeff.q in (5, 10):
                    return tan(pi/2 - arg)
                if pi_coeff.q > 2 and not pi_coeff.q % 2:
                    narg = pi_coeff*pi*2
                    cresult, sresult = cos(narg), cos(narg - pi/2)
                    if not isinstance(cresult, cos) \
                            and not isinstance(sresult, cos):
                        return 1/sresult + cresult/sresult
                q = pi_coeff.q
                p = pi_coeff.p % q
                table2 = _table2()
                if q in table2:
                    a, b = table2[q]
                    nvala, nvalb = cls(p*pi/a), cls(p*pi/b)
                    if None in (nvala, nvalb):
                        return None
                    return (1 + nvala*nvalb)/(nvalb - nvala)
                narg = (((pi_coeff + S.Half) % 1) - S.Half)*pi
                # see cos() to specify which expressions should be
                # expanded automatically in terms of radicals
                cresult, sresult = cos(narg), cos(narg - pi/2)
                if not isinstance(cresult, cos) \
                        and not isinstance(sresult, cos):
                    if sresult == 0:
                        return S.ComplexInfinity
                    return cresult/sresult
                if narg != arg:
                    return cls(narg)

        if arg.is_Add:
            x, m = _peeloff_pi(arg)
            if m:
                cotm = cot(m*pi)
                if cotm is S.ComplexInfinity:
                    return cot(x)
                else: # cotm == 0
                    return -tan(x)

        if arg.is_zero:
            return S.ComplexInfinity

        if isinstance(arg, acot):
            return arg.args[0]

        if isinstance(arg, atan):
            x = arg.args[0]
            return 1/x

        if isinstance(arg, atan2):
            y, x = arg.args
            return x/y

        if isinstance(arg, asin):
            x = arg.args[0]
            return sqrt(1 - x**2)/x

        if isinstance(arg, acos):
            x = arg.args[0]
            return x/sqrt(1 - x**2)

        if isinstance(arg, acsc):
            x = arg.args[0]
            return sqrt(1 - 1/x**2)*x

        if isinstance(arg, asec):
            x = arg.args[0]
            return 1/(sqrt(1 - 1/x**2)*x)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return 1/sympify(x)
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)

            B = bernoulli(n + 1)
            F = factorial(n + 1)

            return S.NegativeOne**((n + 1)//2)*2**(n + 1)*B/F*x**n

    def _eval_nseries(self, x, n, logx, cdir=0):
        i = self.args[0].limit(x, 0)/pi
        if i and i.is_Integer:
            return self.rewrite(cos)._eval_nseries(x, n=n, logx=logx)
        return self.rewrite(tan)._eval_nseries(x, n=n, logx=logx)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        re, im = self._as_real_imag(deep=deep, **hints)
        if im:
            from sympy.functions.elementary.hyperbolic import cosh, sinh
            denom = cos(2*re) - cosh(2*im)
            return (-sin(2*re)/denom, sinh(2*im)/denom)
        else:
            return (self.func(re), S.Zero)

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        from sympy.functions.elementary.hyperbolic import HyperbolicFunction
        I = S.ImaginaryUnit
        if isinstance(arg, (TrigonometricFunction, HyperbolicFunction)):
            arg = arg.func(arg.args[0]).rewrite(exp, **kwargs)
        neg_exp, pos_exp = exp(-arg*I), exp(arg*I)
        return I*(pos_exp + neg_exp)/(pos_exp - neg_exp)

    def _eval_rewrite_as_Pow(self, arg, **kwargs):
        if isinstance(arg, log):
            I = S.ImaginaryUnit
            x = arg.args[0]
            return -I*(x**-I + x**I)/(x**-I - x**I)

    def _eval_rewrite_as_sin(self, x, **kwargs):
        return sin(2*x)/(2*(sin(x)**2))

    def _eval_rewrite_as_cos(self, x, **kwargs):
        return cos(x)/cos(x - pi/2, evaluate=False)

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return cos(arg)/sin(arg)

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        return 1/tan(arg)

    def _eval_rewrite_as_sec(self, arg, **kwargs):
        cos_in_sec_form = cos(arg).rewrite(sec, **kwargs)
        sin_in_sec_form = sin(arg).rewrite(sec, **kwargs)
        return cos_in_sec_form/sin_in_sec_form

    def _eval_rewrite_as_csc(self, arg, **kwargs):
        cos_in_csc_form = cos(arg).rewrite(csc, **kwargs)
        sin_in_csc_form = sin(arg).rewrite(csc, **kwargs)
        return cos_in_csc_form/sin_in_csc_form

    def _eval_rewrite_as_pow(self, arg, **kwargs):
        y = self.rewrite(cos, **kwargs).rewrite(pow, **kwargs)
        if y.has(cos):
            return None
        return y

    def _eval_rewrite_as_sqrt(self, arg, **kwargs):
        y = self.rewrite(cos, **kwargs).rewrite(sqrt, **kwargs)
        if y.has(cos):
            return None
        return y

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return besselj(-S.Half, arg)/besselj(S.Half, arg)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.functions.elementary.complexes import re
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = 2*x0/pi
        if n.is_integer:
            lt = (arg - n*pi/2).as_leading_term(x)
            return 1/lt if n.is_even else -lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in (S.Infinity, S.NegativeInfinity):
            return AccumBounds(S.NegativeInfinity, S.Infinity)
        return self.func(x0) if x0.is_finite else self

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real

    def _eval_expand_trig(self, **hints):
        arg = self.args[0]
        x = None
        if arg.is_Add:
            n = len(arg.args)
            CX = []
            for x in arg.args:
                cx = cot(x, evaluate=False)._eval_expand_trig()
                CX.append(cx)

            Yg = numbered_symbols('Y')
            Y = [ next(Yg) for i in range(n) ]

            p = [0, 0]
            for i in range(n, -1, -1):
                p[(n - i) % 2] += symmetric_poly(i, Y)*(-1)**(((n - i) % 4)//2)
            return (p[0]/p[1]).subs(list(zip(Y, CX)))
        elif arg.is_Mul:
            coeff, terms = arg.as_coeff_Mul(rational=True)
            if coeff.is_Integer and coeff > 1:
                I = S.ImaginaryUnit
                z = Symbol('dummy', real=True)
                P = ((z + I)**coeff).expand()
                return (re(P)/im(P)).subs([(z, cot(terms))])
        return cot(arg)  # XXX sec and csc return 1/cos and 1/sin

    def _eval_is_finite(self):
        arg = self.args[0]
        if arg.is_real and (arg/pi).is_integer is False:
            return True
        if arg.is_imaginary:
            return True

    def _eval_is_real(self):
        arg = self.args[0]
        if arg.is_real and (arg/pi).is_integer is False:
            return True

    def _eval_is_complex(self):
        arg = self.args[0]
        if arg.is_real and (arg/pi).is_integer is False:
            return True

    def _eval_is_zero(self):
        rest, pimult = _peeloff_pi(self.args[0])
        if pimult and rest.is_zero:
            return (pimult - S.Half).is_integer

    def _eval_subs(self, old, new):
        arg = self.args[0]
        argnew = arg.subs(old, new)
        if arg != argnew and (argnew/pi).is_integer:
            return S.ComplexInfinity
        return cot(argnew)


class ReciprocalTrigonometricFunction(TrigonometricFunction):
    """Base class for reciprocal functions of trigonometric functions. """

    _reciprocal_of = None       # mandatory, to be defined in subclass
    _singularities = (S.ComplexInfinity,)

    # _is_even and _is_odd are used for correct evaluation of csc(-x), sec(-x)
    # TODO refactor into TrigonometricFunction common parts of
    # trigonometric functions eval() like even/odd, func(x+2*k*pi), etc.

    # optional, to be defined in subclasses:
    _is_even: FuzzyBool = None
    _is_odd: FuzzyBool = None

    @classmethod
    def eval(cls, arg):
        if arg.could_extract_minus_sign():
            if cls._is_even:
                return cls(-arg)
            if cls._is_odd:
                return -cls(-arg)

        pi_coeff = _pi_coeff(arg)
        if (pi_coeff is not None
            and not (2*pi_coeff).is_integer
            and pi_coeff.is_Rational):
                q = pi_coeff.q
                p = pi_coeff.p % (2*q)
                if p > q:
                    narg = (pi_coeff - 1)*pi
                    return -cls(narg)
                if 2*p > q:
                    narg = (1 - pi_coeff)*pi
                    if cls._is_odd:
                        return cls(narg)
                    elif cls._is_even:
                        return -cls(narg)

        if hasattr(arg, 'inverse') and arg.inverse() == cls:
            return arg.args[0]

        t = cls._reciprocal_of.eval(arg)
        if t is None:
            return t
        elif any(isinstance(i, cos) for i in (t, -t)):
            return (1/t).rewrite(sec)
        elif any(isinstance(i, sin) for i in (t, -t)):
            return (1/t).rewrite(csc)
        else:
            return 1/t

    def _call_reciprocal(self, method_name, *args, **kwargs):
        # Calls method_name on _reciprocal_of
        o = self._reciprocal_of(self.args[0])
        return getattr(o, method_name)(*args, **kwargs)

    def _calculate_reciprocal(self, method_name, *args, **kwargs):
        # If calling method_name on _reciprocal_of returns a value != None
        # then return the reciprocal of that value
        t = self._call_reciprocal(method_name, *args, **kwargs)
        return 1/t if t is not None else t

    def _rewrite_reciprocal(self, method_name, arg):
        # Special handling for rewrite functions. If reciprocal rewrite returns
        # unmodified expression, then return None
        t = self._call_reciprocal(method_name, arg)
        if t is not None and t != self._reciprocal_of(arg):
            return 1/t

    def _period(self, symbol):
        f = expand_mul(self.args[0])
        return self._reciprocal_of(f).period(symbol)

    def fdiff(self, argindex=1):
        return -self._calculate_reciprocal("fdiff", argindex)/self**2

    def _eval_rewrite_as_exp(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_exp", arg)

    def _eval_rewrite_as_Pow(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_Pow", arg)

    def _eval_rewrite_as_sin(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_sin", arg)

    def _eval_rewrite_as_cos(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_cos", arg)

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_tan", arg)

    def _eval_rewrite_as_pow(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_pow", arg)

    def _eval_rewrite_as_sqrt(self, arg, **kwargs):
        return self._rewrite_reciprocal("_eval_rewrite_as_sqrt", arg)

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate())

    def as_real_imag(self, deep=True, **hints):
        return (1/self._reciprocal_of(self.args[0])).as_real_imag(deep,
                                                                  **hints)

    def _eval_expand_trig(self, **hints):
        return self._calculate_reciprocal("_eval_expand_trig", **hints)

    def _eval_is_extended_real(self):
        return self._reciprocal_of(self.args[0])._eval_is_extended_real()

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        return (1/self._reciprocal_of(self.args[0]))._eval_as_leading_term(x)

    def _eval_is_finite(self):
        return (1/self._reciprocal_of(self.args[0])).is_finite

    def _eval_nseries(self, x, n, logx, cdir=0):
        return (1/self._reciprocal_of(self.args[0]))._eval_nseries(x, n, logx)


class sec(ReciprocalTrigonometricFunction):
    """
    The secant function.

    Returns the secant of x (measured in radians).

    Explanation
    ===========

    See :class:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import sec
    >>> from sympy.abc import x
    >>> sec(x**2).diff(x)
    2*x*tan(x**2)*sec(x**2)
    >>> sec(1).diff(x)
    0

    See Also
    ========

    sin, csc, cos, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Sec

    """

    _reciprocal_of = cos
    _is_even = True

    def period(self, symbol=None):
        return self._period(symbol)

    def _eval_rewrite_as_cot(self, arg, **kwargs):
        cot_half_sq = cot(arg/2)**2
        return (cot_half_sq + 1)/(cot_half_sq - 1)

    def _eval_rewrite_as_cos(self, arg, **kwargs):
        return (1/cos(arg))

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return sin(arg)/(cos(arg)*sin(arg))

    def _eval_rewrite_as_sin(self, arg, **kwargs):
        return (1/cos(arg).rewrite(sin, **kwargs))

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        return (1/cos(arg).rewrite(tan, **kwargs))

    def _eval_rewrite_as_csc(self, arg, **kwargs):
        return csc(pi/2 - arg, evaluate=False)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return tan(self.args[0])*sec(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return Piecewise(
                (1/(sqrt(pi*arg)/(sqrt(2))*besselj(-S.Half, arg)), Ne(arg, 0)),
                (1, True)
            )

    def _eval_is_complex(self):
        arg = self.args[0]

        if arg.is_complex and (arg/pi - S.Half).is_integer is False:
            return True

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        # Reference Formula:
        # https://functions.wolfram.com/ElementaryFunctions/Sec/06/01/02/01/
        if n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)
            k = n//2
            return S.NegativeOne**k*euler(2*k)/factorial(2*k)*x**(2*k)

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.functions.elementary.complexes import re
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = (x0 + pi/2)/pi
        if n.is_integer:
            lt = (arg - n*pi + pi/2).as_leading_term(x)
            return (S.NegativeOne**n)/lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in (S.Infinity, S.NegativeInfinity):
            return AccumBounds(S.NegativeInfinity, S.Infinity)
        return self.func(x0) if x0.is_finite else self


class csc(ReciprocalTrigonometricFunction):
    """
    The cosecant function.

    Returns the cosecant of x (measured in radians).

    Explanation
    ===========

    See :func:`sin` for notes about automatic evaluation.

    Examples
    ========

    >>> from sympy import csc
    >>> from sympy.abc import x
    >>> csc(x**2).diff(x)
    -2*x*cot(x**2)*csc(x**2)
    >>> csc(1).diff(x)
    0

    See Also
    ========

    sin, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.14
    .. [3] https://functions.wolfram.com/ElementaryFunctions/Csc

    """

    _reciprocal_of = sin
    _is_odd = True

    def period(self, symbol=None):
        return self._period(symbol)

    def _eval_rewrite_as_sin(self, arg, **kwargs):
        return (1/sin(arg))

    def _eval_rewrite_as_sincos(self, arg, **kwargs):
        return cos(arg)/(sin(arg)*cos(arg))

    def _eval_rewrite_as_cot(self, arg, **kwargs):
        cot_half = cot(arg/2)
        return (1 + cot_half**2)/(2*cot_half)

    def _eval_rewrite_as_cos(self, arg, **kwargs):
        return 1/sin(arg).rewrite(cos, **kwargs)

    def _eval_rewrite_as_sec(self, arg, **kwargs):
        return sec(pi/2 - arg, evaluate=False)

    def _eval_rewrite_as_tan(self, arg, **kwargs):
        return (1/sin(arg).rewrite(tan, **kwargs))

    def _eval_rewrite_as_besselj(self, arg, **kwargs):
        from sympy.functions.special.bessel import besselj
        return sqrt(2/pi)*(1/(sqrt(arg)*besselj(S.Half, arg)))

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -cot(self.args[0])*csc(self.args[0])
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_complex(self):
        arg = self.args[0]
        if arg.is_real and (arg/pi).is_integer is False:
            return True

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return 1/sympify(x)
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            k = n//2 + 1
            return (S.NegativeOne**(k - 1)*2*(2**(2*k - 1) - 1)*
                    bernoulli(2*k)*x**(2*k - 1)/factorial(2*k))

    def _eval_as_leading_term(self, x, logx=None, cdir=0):
        from sympy.calculus.accumulationbounds import AccumBounds
        from sympy.functions.elementary.complexes import re
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        n = x0/pi
        if n.is_integer:
            lt = (arg - n*pi).as_leading_term(x)
            return (S.NegativeOne**n)/lt
        if x0 is S.ComplexInfinity:
            x0 = arg.limit(x, 0, dir='-' if re(cdir).is_negative else '+')
        if x0 in (S.Infinity, S.NegativeInfinity):
            return AccumBounds(S.NegativeInfinity, S.Infinity)
        return self.func(x0) if x0.is_finite else self


class sinc(Function):
    r"""
    Represents an unnormalized sinc function:

    .. math::

        \operatorname{sinc}(x) =
        \begin{cases}
          \frac{\sin x}{x} & \qquad x \neq 0 \\
          1 & \qquad x = 0
        \end{cases}

    Examples
    ========

    >>> from sympy import sinc, oo, jn
    >>> from sympy.abc import x
    >>> sinc(x)
    sinc(x)

    * Automated Evaluation

    >>> sinc(0)
    1
    >>> sinc(oo)
    0

    * Differentiation

    >>> sinc(x).diff()
    cos(x)/x - sin(x)/x**2

    * Series Expansion

    >>> sinc(x).series()
    1 - x**2/6 + x**4/120 + O(x**6)

    * As zero'th order spherical Bessel Function

    >>> sinc(x).rewrite(jn)
    jn(0, x)

    See also
    ========

    sin

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Sinc_function

    """
    _singularities = (S.ComplexInfinity,)

    def fdiff(self, argindex=1):
        x = self.args[0]
        if argindex == 1:
            # We would like to return the Piecewise here, but Piecewise.diff
            # currently can't handle removable singularities, meaning things
            # like sinc(x).diff(x, 2) give the wrong answer at x = 0. See
            # https://github.com/sympy/sympy/issues/11402.
            #
            # return Piecewise(((x*cos(x) - sin(x))/x**2, Ne(x, S.Zero)), (S.Zero, S.true))
            return cos(x)/x - sin(x)/x**2
        else:
            raise ArgumentIndexError(self, argindex)

    @classmethod
    def eval(cls, arg):
        if arg.is_zero:
            return S.One
        if arg.is_Number:
            if arg in [S.Infinity, S.NegativeInfinity]:
                return S.Zero
            elif arg is S.NaN:
                return S.NaN

        if arg is S.ComplexInfinity:
            return S.NaN

        if arg.could_extract_minus_sign():
            return cls(-arg)

        pi_coeff = _pi_coeff(arg)
        if pi_coeff is not None:
            if pi_coeff.is_integer:
                if fuzzy_not(arg.is_zero):
                    return S.Zero
            elif (2*pi_coeff).is_integer:
                return S.NegativeOne**(pi_coeff - S.Half)/arg

    def _eval_nseries(self, x, n, logx, cdir=0):
        x = self.args[0]
        return (sin(x)/x)._eval_nseries(x, n, logx)

    def _eval_rewrite_as_jn(self, arg, **kwargs):
        from sympy.functions.special.bessel import jn
        return jn(0, arg)

    def _eval_rewrite_as_sin(self, arg, **kwargs):
        return Piecewise((sin(arg)/arg, Ne(arg, S.Zero)), (S.One, S.true))

    def _eval_is_zero(self):
        if self.args[0].is_infinite:
            return True
        rest, pi_mult = _peeloff_pi(self.args[0])
        if rest.is_zero:
            return fuzzy_and([pi_mult.is_integer, pi_mult.is_nonzero])
        if rest.is_Number and pi_mult.is_integer:
            return False

    def _eval_is_real(self):
        if self.args[0].is_extended_real or self.args[0].is_imaginary:
            return True

    _eval_is_finite = _eval_is_real


###############################################################################
########################### TRIGONOMETRIC INVERSES ############################
###############################################################################


class InverseTrigonometricFunction(Function):
    """Base class for inverse trigonometric functions."""
    _singularities = (S.One, S.NegativeOne, S.Zero, S.ComplexInfinity)  # type: tTuple[Expr, ...]

    @staticmethod
    @cacheit
    def _asin_table():
        # Only keys with could_extract_minus_sign() == False
        # are actually needed.
        return {
            sqrt(3)/2: pi/3,
            sqrt(2)/2: pi/4,
            1/sqrt(2): pi/4,
            sqrt((5 - sqrt(5))/8): pi/5,
            sqrt(2)*sqrt(5 - sqrt(5))/4: pi/5,
            sqrt((5 + sqrt(5))/8): pi*Rational(2, 5),
            sqrt(2)*sqrt(5 + sqrt(5))/4: pi*Rational(2, 5),
            S.Half: pi/6,
            sqrt(2 - sqrt(2))/2: pi/8,
            sqrt(S.Half - sqrt(2)/4): pi/8,
            sqrt(2 + sqrt(2))/2: pi*Rational(3, 8),
            sqrt(S.Half + sqrt(2)/4): pi*Rational(3, 8),
            (sqrt(5) - 1)/4: pi/10,
            (1 - sqrt(5))/4: -pi/10,
            (sqrt(5) + 1)/4: pi*Rational(3, 10),
            sqrt(6)/4 - sqrt(2)/4: pi/12,
            -sqrt(6)/4 + sqrt(2)/4: -pi/12,
            (sqrt(3) - 1)/sqrt(8): pi/12,
            (1 - sqrt(3))/sqrt(8): -pi/12,
            sqrt(6)/4 + sqrt(2)/4: pi*Rational(5, 12),
            (1 + sqrt(3))/sqrt(8): pi*Rational(5, 12)
        }


    @staticmethod
    @cacheit
    def _atan_table():
        # Only keys with could_extract_minus_sign() == False
        # are actually needed.
        return {
            sqrt(3)/3: pi/6,
            1/sqrt(3): pi/6,
            sqrt(3): pi/3,
            sqrt(2) - 1: pi/8,
            1 - sqrt(2): -pi/8,
            1 + sqrt(2): pi*Rational(3, 8),
            sqrt(5 - 2*sqrt(5)): pi/5,
            sqrt(5 + 2*sqrt(5)): pi*Rational(2, 5),
            sqrt(1 - 2*sqrt(5)/5): pi/10,
            sqrt(1 + 2*sqrt(5)/5): pi*Rational(3, 10),
            2 - sqrt(3): pi/12,
            -2 + sqrt(3): -pi/12,
            2 + sqrt(3): pi*Rational(5, 12)
        }

    @staticmethod
    @cacheit
    def _acsc_table():
        # Keys for which could_extract_minus_sign()
        # will obviously return True are omitted.
        return {
            2*sqrt(3)/3: pi/3,
            sqrt(2): pi/4,
            sqrt(2 + 2*sqrt(5)/5): pi/5,
            1/sqrt(Rational(5, 8) - sqrt(5)/8): pi/5,
            sqrt(2 - 2*sqrt(5)/5): pi*Rational(2, 5),
            1/sqrt(Rational(5, 8) + sqrt(5)/8): pi*Rational(2, 5),
            2: pi/6,
            sqrt(4 + 2*sqrt(2)): pi/8,
            2/sqrt(2 - sqrt(2)): pi/8,
            sqrt(4 - 2*sqrt(2)): pi*Rational(3, 8),
            2/sqrt(2 + sqrt(2)): pi*Rational(3, 8),
            1 + sqrt(5): pi/10,
            sqrt(5) - 1: pi*Rational(3, 10),
            -(sqrt(5) - 1): pi*Rational(-3, 10),
            sqrt(6) + sqrt(2): pi/12,
            sqrt(6) - sqrt(2): pi*Rational(5, 12),
            -(sqrt(6) - sqrt(2)): pi*Rational(-5, 12)
        }


class asin(InverseTrigonometricFunction):
    r"""
    The inverse sine function.

    Returns the arcsine of x in radians.

    Explanation
    ===========

    ``asin(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the ``eval`` class method).

    A purely imaginary argument will lead to an asinh expression.

    Examples
    ========

    >>> from sympy import asin, oo
    >>> asin(1)
    pi/2
    >>> asin(-1)
    -pi/2
    >>> asin(-oo)
    oo*I
    >>> asin(oo)
    -oo*I

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    acsc, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSin

    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/sqrt(1 - self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational:
                return False
        else:
            return s.is_rational

    def _eval_is_positive(self):
        return self._eval_is_extended_real() and self.args[0].is_positive

    def _eval_is_negative(self):
        return self._eval_is_extended_real() and self.args[0].is_negative

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.NegativeInfinity*S.ImaginaryUnit
            elif arg is S.NegativeInfinity:
                return S.Infinity*S.ImaginaryUnit
            elif arg.is_zero:
                return S.Zero
            elif arg is S.One:
                return pi/2
            elif arg is S.NegativeOne:
                return -pi/2

        if arg is S.ComplexInfinity:
            return S.ComplexInfinity

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        if arg.is_number:
            asin_table = cls._asin_table()
            if arg in asin_table:
                return asin_table[arg]

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import asinh
            return S.ImaginaryUnit*asinh(i_coeff)

        if arg.is_zero:
            return S.Zero

        if isinstance(arg, sin):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= 2*pi # restrict to [0,2*pi)
                if ang > pi: # restrict to (-pi,pi]
                    ang = pi - ang

                # restrict to [-pi/2,pi/2]
                if ang > pi/2:
                    ang = pi - ang
                if ang < -pi/2:
                    ang = -pi - ang

                return ang

        if isinstance(arg, cos): # acos(x) + asin(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                return pi/2 - acos(arg)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) >= 2 and n > 2:
                p = previous_terms[-2]
                return p*(n - 2)**2/(n*(n - 1))*x**2
            else:
                k = (n - 1) // 2
                R = RisingFactorial(S.Half, k)
                F = factorial(k)
                return R/F*x**n/n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # asin
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        if x0.is_zero:
            return arg.as_leading_term(x)

        # Handling branch points
        if x0 in (-S.One, S.One, S.ComplexInfinity):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        # Handling points lying on branch cuts (-oo, -1) U (1, oo)
        if (1 - x0**2).is_negative:
            ndir = arg.dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if x0.is_negative:
                    return -pi - self.func(x0)
            elif im(ndir).is_positive:
                if x0.is_positive:
                    return pi - self.func(x0)
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_nseries(self, x, n, logx, cdir=0):  # asin
        from sympy.series.order import O
        arg0 = self.args[0].subs(x, 0)
        # Handling branch points
        if arg0 is S.One:
            t = Dummy('t', positive=True)
            ser = asin(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.One - self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            if not g.is_meromorphic(x, 0):   # cannot be expanded
                return O(1) if n == 0 else pi/2 + O(sqrt(x))
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        if arg0 is S.NegativeOne:
            t = Dummy('t', positive=True)
            ser = asin(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.One + self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            if not g.is_meromorphic(x, 0):   # cannot be expanded
                return O(1) if n == 0 else -pi/2 + O(sqrt(x))
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        if arg0 is S.ComplexInfinity:
            return res
        # Handling points lying on branch cuts (-oo, -1) U (1, oo)
        if (1 - arg0**2).is_negative:
            ndir = self.args[0].dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if arg0.is_negative:
                    return -pi - res
            elif im(ndir).is_positive:
                if arg0.is_positive:
                    return pi - res
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_rewrite_as_acos(self, x, **kwargs):
        return pi/2 - acos(x)

    def _eval_rewrite_as_atan(self, x, **kwargs):
        return 2*atan(x/(1 + sqrt(1 - x**2)))

    def _eval_rewrite_as_log(self, x, **kwargs):
        return -S.ImaginaryUnit*log(S.ImaginaryUnit*x + sqrt(1 - x**2))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def _eval_rewrite_as_acot(self, arg, **kwargs):
        return 2*acot((1 + sqrt(1 - arg**2))/arg)

    def _eval_rewrite_as_asec(self, arg, **kwargs):
        return pi/2 - asec(1/arg)

    def _eval_rewrite_as_acsc(self, arg, **kwargs):
        return acsc(1/arg)

    def _eval_is_extended_real(self):
        x = self.args[0]
        return x.is_extended_real and (1 - abs(x)).is_nonnegative

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return sin


class acos(InverseTrigonometricFunction):
    r"""
    The inverse cosine function.

    Explanation
    ===========

    Returns the arc cosine of x (measured in radians).

    ``acos(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when
    the result is a rational multiple of $\pi$ (see the eval class method).

    ``acos(zoo)`` evaluates to ``zoo``
    (see note in :class:`sympy.functions.elementary.trigonometric.asec`)

    A purely imaginary argument will be rewritten to asinh.

    Examples
    ========

    >>> from sympy import acos, oo
    >>> acos(1)
    0
    >>> acos(0)
    pi/2
    >>> acos(oo)
    oo*I

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCos

    """

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -1/sqrt(1 - self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational:
                return False
        else:
            return s.is_rational

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Infinity*S.ImaginaryUnit
            elif arg is S.NegativeInfinity:
                return S.NegativeInfinity*S.ImaginaryUnit
            elif arg.is_zero:
                return pi/2
            elif arg is S.One:
                return S.Zero
            elif arg is S.NegativeOne:
                return pi

        if arg is S.ComplexInfinity:
            return S.ComplexInfinity

        if arg.is_number:
            asin_table = cls._asin_table()
            if arg in asin_table:
                return pi/2 - asin_table[arg]
            elif -arg in asin_table:
                return pi/2 + asin_table[-arg]

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            return pi/2 - asin(arg)

        if isinstance(arg, cos):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= 2*pi # restrict to [0,2*pi)
                if ang > pi: # restrict to [0,pi]
                    ang = 2*pi - ang

                return ang

        if isinstance(arg, sin): # acos(x) + asin(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                return pi/2 - asin(arg)

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return pi/2
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) >= 2 and n > 2:
                p = previous_terms[-2]
                return p*(n - 2)**2/(n*(n - 1))*x**2
            else:
                k = (n - 1) // 2
                R = RisingFactorial(S.Half, k)
                F = factorial(k)
                return -R/F*x**n/n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # acos
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        # Handling branch points
        if x0 == 1:
            return sqrt(2)*sqrt((S.One - arg).as_leading_term(x))
        if x0 in (-S.One, S.ComplexInfinity):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
        # Handling points lying on branch cuts (-oo, -1) U (1, oo)
        if (1 - x0**2).is_negative:
            ndir = arg.dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if x0.is_negative:
                    return 2*pi - self.func(x0)
            elif im(ndir).is_positive:
                if x0.is_positive:
                    return -self.func(x0)
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_is_extended_real(self):
        x = self.args[0]
        return x.is_extended_real and (1 - abs(x)).is_nonnegative

    def _eval_is_nonnegative(self):
        return self._eval_is_extended_real()

    def _eval_nseries(self, x, n, logx, cdir=0):  # acos
        from sympy.series.order import O
        arg0 = self.args[0].subs(x, 0)
        # Handling branch points
        if arg0 is S.One:
            t = Dummy('t', positive=True)
            ser = acos(S.One - t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.One - self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            if not g.is_meromorphic(x, 0):   # cannot be expanded
                return O(1) if n == 0 else O(sqrt(x))
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        if arg0 is S.NegativeOne:
            t = Dummy('t', positive=True)
            ser = acos(S.NegativeOne + t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.One + self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            if not g.is_meromorphic(x, 0):   # cannot be expanded
                return O(1) if n == 0 else pi + O(sqrt(x))
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        if arg0 is S.ComplexInfinity:
            return res
        # Handling points lying on branch cuts (-oo, -1) U (1, oo)
        if (1 - arg0**2).is_negative:
            ndir = self.args[0].dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if arg0.is_negative:
                    return 2*pi - res
            elif im(ndir).is_positive:
                if arg0.is_positive:
                    return -res
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_rewrite_as_log(self, x, **kwargs):
        return pi/2 + S.ImaginaryUnit*\
            log(S.ImaginaryUnit*x + sqrt(1 - x**2))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def _eval_rewrite_as_asin(self, x, **kwargs):
        return pi/2 - asin(x)

    def _eval_rewrite_as_atan(self, x, **kwargs):
        return atan(sqrt(1 - x**2)/x) + (pi/2)*(1 - x*sqrt(1/x**2))

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return cos

    def _eval_rewrite_as_acot(self, arg, **kwargs):
        return pi/2 - 2*acot((1 + sqrt(1 - arg**2))/arg)

    def _eval_rewrite_as_asec(self, arg, **kwargs):
        return asec(1/arg)

    def _eval_rewrite_as_acsc(self, arg, **kwargs):
        return pi/2 - acsc(1/arg)

    def _eval_conjugate(self):
        z = self.args[0]
        r = self.func(self.args[0].conjugate())
        if z.is_extended_real is False:
            return r
        elif z.is_extended_real and (z + 1).is_nonnegative and (z - 1).is_nonpositive:
            return r


class atan(InverseTrigonometricFunction):
    r"""
    The inverse tangent function.

    Returns the arc tangent of x (measured in radians).

    Explanation
    ===========

    ``atan(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the eval class method).

    Examples
    ========

    >>> from sympy import atan, oo
    >>> atan(0)
    0
    >>> atan(1)
    pi/4
    >>> atan(oo)
    pi/2

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan

    """

    args: tTuple[Expr]

    _singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/(1 + self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational:
                return False
        else:
            return s.is_rational

    def _eval_is_positive(self):
        return self.args[0].is_extended_positive

    def _eval_is_nonnegative(self):
        return self.args[0].is_extended_nonnegative

    def _eval_is_zero(self):
        return self.args[0].is_zero

    def _eval_is_real(self):
        return self.args[0].is_extended_real

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return pi/2
            elif arg is S.NegativeInfinity:
                return -pi/2
            elif arg.is_zero:
                return S.Zero
            elif arg is S.One:
                return pi/4
            elif arg is S.NegativeOne:
                return -pi/4

        if arg is S.ComplexInfinity:
            from sympy.calculus.accumulationbounds import AccumBounds
            return AccumBounds(-pi/2, pi/2)

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        if arg.is_number:
            atan_table = cls._atan_table()
            if arg in atan_table:
                return atan_table[arg]

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import atanh
            return S.ImaginaryUnit*atanh(i_coeff)

        if arg.is_zero:
            return S.Zero

        if isinstance(arg, tan):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= pi # restrict to [0,pi)
                if ang > pi/2: # restrict to [-pi/2,pi/2]
                    ang -= pi

                return ang

        if isinstance(arg, cot): # atan(x) + acot(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                ang = pi/2 - acot(arg)
                if ang > pi/2: # restrict to [-pi/2,pi/2]
                    ang -= pi
                return ang

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            return S.NegativeOne**((n - 1)//2)*x**n/n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # atan
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        if x0.is_zero:
            return arg.as_leading_term(x)
        # Handling branch points
        if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.ComplexInfinity):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        # Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
        if (1 + x0**2).is_negative:
            ndir = arg.dir(x, cdir if cdir else 1)
            if re(ndir).is_negative:
                if im(x0).is_positive:
                    return self.func(x0) - pi
            elif re(ndir).is_positive:
                if im(x0).is_negative:
                    return self.func(x0) + pi
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_nseries(self, x, n, logx, cdir=0):  # atan
        arg0 = self.args[0].subs(x, 0)

        # Handling branch points
        if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
            return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        ndir = self.args[0].dir(x, cdir if cdir else 1)
        if arg0 is S.ComplexInfinity:
            if re(ndir) > 0:
                return res - pi
            return res
        # Handling points lying on branch cuts (-I*oo, -I) U (I, I*oo)
        if (1 + arg0**2).is_negative:
            if re(ndir).is_negative:
                if im(arg0).is_positive:
                    return res - pi
            elif re(ndir).is_positive:
                if im(arg0).is_negative:
                    return res + pi
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_rewrite_as_log(self, x, **kwargs):
        return S.ImaginaryUnit/2*(log(S.One - S.ImaginaryUnit*x)
            - log(S.One + S.ImaginaryUnit*x))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def _eval_aseries(self, n, args0, x, logx):
        if args0[0] in [S.Infinity, S.NegativeInfinity]:
            return (pi/2 - atan(1/self.args[0]))._eval_nseries(x, n, logx)
        else:
            return super()._eval_aseries(n, args0, x, logx)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return tan

    def _eval_rewrite_as_asin(self, arg, **kwargs):
        return sqrt(arg**2)/arg*(pi/2 - asin(1/sqrt(1 + arg**2)))

    def _eval_rewrite_as_acos(self, arg, **kwargs):
        return sqrt(arg**2)/arg*acos(1/sqrt(1 + arg**2))

    def _eval_rewrite_as_acot(self, arg, **kwargs):
        return acot(1/arg)

    def _eval_rewrite_as_asec(self, arg, **kwargs):
        return sqrt(arg**2)/arg*asec(sqrt(1 + arg**2))

    def _eval_rewrite_as_acsc(self, arg, **kwargs):
        return sqrt(arg**2)/arg*(pi/2 - acsc(sqrt(1 + arg**2)))


class acot(InverseTrigonometricFunction):
    r"""
    The inverse cotangent function.

    Returns the arc cotangent of x (measured in radians).

    Explanation
    ===========

    ``acot(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, \tilde{\infty}, 0, 1, -1\}$
    and for some instances when the result is a rational multiple of $\pi$
    (see the eval class method).

    A purely imaginary argument will lead to an ``acoth`` expression.

    ``acot(x)`` has a branch cut along $(-i, i)$, hence it is discontinuous
    at 0. Its range for real $x$ is $(-\frac{\pi}{2}, \frac{\pi}{2}]$.

    Examples
    ========

    >>> from sympy import acot, sqrt
    >>> acot(0)
    pi/2
    >>> acot(1)
    pi/4
    >>> acot(sqrt(3) - 2)
    -5*pi/12

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, atan2

    References
    ==========

    .. [1] https://dlmf.nist.gov/4.23
    .. [2] https://functions.wolfram.com/ElementaryFunctions/ArcCot

    """
    _singularities = (S.ImaginaryUnit, -S.ImaginaryUnit)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -1/(1 + self.args[0]**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_is_rational(self):
        s = self.func(*self.args)
        if s.func == self.func:
            if s.args[0].is_rational:
                return False
        else:
            return s.is_rational

    def _eval_is_positive(self):
        return self.args[0].is_nonnegative

    def _eval_is_negative(self):
        return self.args[0].is_negative

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real

    @classmethod
    def eval(cls, arg):
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.Infinity:
                return S.Zero
            elif arg is S.NegativeInfinity:
                return S.Zero
            elif arg.is_zero:
                return pi/ 2
            elif arg is S.One:
                return pi/4
            elif arg is S.NegativeOne:
                return -pi/4

        if arg is S.ComplexInfinity:
            return S.Zero

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        if arg.is_number:
            atan_table = cls._atan_table()
            if arg in atan_table:
                ang = pi/2 - atan_table[arg]
                if ang > pi/2: # restrict to (-pi/2,pi/2]
                    ang -= pi
                return ang

        i_coeff = _imaginary_unit_as_coefficient(arg)
        if i_coeff is not None:
            from sympy.functions.elementary.hyperbolic import acoth
            return -S.ImaginaryUnit*acoth(i_coeff)

        if arg.is_zero:
            return pi*S.Half

        if isinstance(arg, cot):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= pi # restrict to [0,pi)
                if ang > pi/2: # restrict to (-pi/2,pi/2]
                    ang -= pi;
                return ang

        if isinstance(arg, tan): # atan(x) + acot(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                ang = pi/2 - atan(arg)
                if ang > pi/2: # restrict to (-pi/2,pi/2]
                    ang -= pi
                return ang

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return pi/2  # FIX THIS
        elif n < 0 or n % 2 == 0:
            return S.Zero
        else:
            x = sympify(x)
            return S.NegativeOne**((n + 1)//2)*x**n/n

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # acot
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        if x0 is S.ComplexInfinity:
            return (1/arg).as_leading_term(x)
        # Handling branch points
        if x0 in (-S.ImaginaryUnit, S.ImaginaryUnit, S.Zero):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        # Handling points lying on branch cuts [-I, I]
        if x0.is_imaginary and (1 + x0**2).is_positive:
            ndir = arg.dir(x, cdir if cdir else 1)
            if re(ndir).is_positive:
                if im(x0).is_positive:
                    return self.func(x0) + pi
            elif re(ndir).is_negative:
                if im(x0).is_negative:
                    return self.func(x0) - pi
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_nseries(self, x, n, logx, cdir=0):  # acot
        arg0 = self.args[0].subs(x, 0)

        # Handling branch points
        if arg0 in (S.ImaginaryUnit, S.NegativeOne*S.ImaginaryUnit):
            return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        if arg0 is S.ComplexInfinity:
            return res
        ndir = self.args[0].dir(x, cdir if cdir else 1)
        if arg0.is_zero:
            if re(ndir) < 0:
                return res - pi
            return res
        # Handling points lying on branch cuts [-I, I]
        if arg0.is_imaginary and (1 + arg0**2).is_positive:
            if re(ndir).is_positive:
                if im(arg0).is_positive:
                    return res + pi
            elif re(ndir).is_negative:
                if im(arg0).is_negative:
                    return res - pi
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_aseries(self, n, args0, x, logx):
        if args0[0] in [S.Infinity, S.NegativeInfinity]:
            return atan(1/self.args[0])._eval_nseries(x, n, logx)
        else:
            return super()._eval_aseries(n, args0, x, logx)

    def _eval_rewrite_as_log(self, x, **kwargs):
        return S.ImaginaryUnit/2*(log(1 - S.ImaginaryUnit/x)
            - log(1 + S.ImaginaryUnit/x))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return cot

    def _eval_rewrite_as_asin(self, arg, **kwargs):
        return (arg*sqrt(1/arg**2)*
                (pi/2 - asin(sqrt(-arg**2)/sqrt(-arg**2 - 1))))

    def _eval_rewrite_as_acos(self, arg, **kwargs):
        return arg*sqrt(1/arg**2)*acos(sqrt(-arg**2)/sqrt(-arg**2 - 1))

    def _eval_rewrite_as_atan(self, arg, **kwargs):
        return atan(1/arg)

    def _eval_rewrite_as_asec(self, arg, **kwargs):
        return arg*sqrt(1/arg**2)*asec(sqrt((1 + arg**2)/arg**2))

    def _eval_rewrite_as_acsc(self, arg, **kwargs):
        return arg*sqrt(1/arg**2)*(pi/2 - acsc(sqrt((1 + arg**2)/arg**2)))


class asec(InverseTrigonometricFunction):
    r"""
    The inverse secant function.

    Returns the arc secant of x (measured in radians).

    Explanation
    ===========

    ``asec(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$ and for some instances when the
    result is a rational multiple of $\pi$ (see the eval class method).

    ``asec(x)`` has branch cut in the interval $[-1, 1]$. For complex arguments,
    it can be defined [4]_ as

    .. math::
        \operatorname{sec^{-1}}(z) = -i\frac{\log\left(\sqrt{1 - z^2} + 1\right)}{z}

    At ``x = 0``, for positive branch cut, the limit evaluates to ``zoo``. For
    negative branch cut, the limit

    .. math::
        \lim_{z \to 0}-i\frac{\log\left(-\sqrt{1 - z^2} + 1\right)}{z}

    simplifies to :math:`-i\log\left(z/2 + O\left(z^3\right)\right)` which
    ultimately evaluates to ``zoo``.

    As ``acos(x) = asec(1/x)``, a similar argument can be given for
    ``acos(x)``.

    Examples
    ========

    >>> from sympy import asec, oo
    >>> asec(1)
    0
    >>> asec(-1)
    pi
    >>> asec(0)
    zoo
    >>> asec(-oo)
    pi/2

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcSec
    .. [4] https://reference.wolfram.com/language/ref/ArcSec.html

    """

    @classmethod
    def eval(cls, arg):
        if arg.is_zero:
            return S.ComplexInfinity
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.One:
                return S.Zero
            elif arg is S.NegativeOne:
                return pi
        if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
            return pi/2

        if arg.is_number:
            acsc_table = cls._acsc_table()
            if arg in acsc_table:
                return pi/2 - acsc_table[arg]
            elif -arg in acsc_table:
                return pi/2 + acsc_table[-arg]

        if arg.is_infinite:
            return pi/2

        if isinstance(arg, sec):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= 2*pi # restrict to [0,2*pi)
                if ang > pi: # restrict to [0,pi]
                    ang = 2*pi - ang

                return ang

        if isinstance(arg, csc): # asec(x) + acsc(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                return pi/2 - acsc(arg)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return 1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return sec

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return S.ImaginaryUnit*log(2 / x)
        elif n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) > 2 and n > 2:
                p = previous_terms[-2]
                return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
            else:
                k = n // 2
                R = RisingFactorial(S.Half, k) *  n
                F = factorial(k) * n // 2 * n // 2
                return -S.ImaginaryUnit * R / F * x**n / 4

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # asec
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        # Handling branch points
        if x0 == 1:
            return sqrt(2)*sqrt((arg - S.One).as_leading_term(x))
        if x0 in (-S.One, S.Zero):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir)
        # Handling points lying on branch cuts (-1, 1)
        if x0.is_real and (1 - x0**2).is_positive:
            ndir = arg.dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if x0.is_positive:
                    return -self.func(x0)
            elif im(ndir).is_positive:
                if x0.is_negative:
                    return 2*pi - self.func(x0)
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_nseries(self, x, n, logx, cdir=0):  # asec
        from sympy.series.order import O
        arg0 = self.args[0].subs(x, 0)
        # Handling branch points
        if arg0 is S.One:
            t = Dummy('t', positive=True)
            ser = asec(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.NegativeOne + self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        if arg0 is S.NegativeOne:
            t = Dummy('t', positive=True)
            ser = asec(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.NegativeOne - self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        if arg0 is S.ComplexInfinity:
            return res
        # Handling points lying on branch cuts (-1, 1)
        if arg0.is_real and (1 - arg0**2).is_positive:
            ndir = self.args[0].dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if arg0.is_positive:
                    return -res
            elif im(ndir).is_positive:
                if arg0.is_negative:
                    return 2*pi - res
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_is_extended_real(self):
        x = self.args[0]
        if x.is_extended_real is False:
            return False
        return fuzzy_or(((x - 1).is_nonnegative, (-x - 1).is_nonnegative))

    def _eval_rewrite_as_log(self, arg, **kwargs):
        return pi/2 + S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def _eval_rewrite_as_asin(self, arg, **kwargs):
        return pi/2 - asin(1/arg)

    def _eval_rewrite_as_acos(self, arg, **kwargs):
        return acos(1/arg)

    def _eval_rewrite_as_atan(self, x, **kwargs):
        sx2x = sqrt(x**2)/x
        return pi/2*(1 - sx2x) + sx2x*atan(sqrt(x**2 - 1))

    def _eval_rewrite_as_acot(self, x, **kwargs):
        sx2x = sqrt(x**2)/x
        return pi/2*(1 - sx2x) + sx2x*acot(1/sqrt(x**2 - 1))

    def _eval_rewrite_as_acsc(self, arg, **kwargs):
        return pi/2 - acsc(arg)


class acsc(InverseTrigonometricFunction):
    r"""
    The inverse cosecant function.

    Returns the arc cosecant of x (measured in radians).

    Explanation
    ===========

    ``acsc(x)`` will evaluate automatically in the cases
    $x \in \{\infty, -\infty, 0, 1, -1\}$` and for some instances when the
    result is a rational multiple of $\pi$ (see the ``eval`` class method).

    Examples
    ========

    >>> from sympy import acsc, oo
    >>> acsc(1)
    pi/2
    >>> acsc(-1)
    -pi/2
    >>> acsc(oo)
    0
    >>> acsc(-oo) == acsc(oo)
    True
    >>> acsc(0)
    zoo

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acos, asec, atan, acot, atan2

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://dlmf.nist.gov/4.23
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcCsc

    """

    @classmethod
    def eval(cls, arg):
        if arg.is_zero:
            return S.ComplexInfinity
        if arg.is_Number:
            if arg is S.NaN:
                return S.NaN
            elif arg is S.One:
                return pi/2
            elif arg is S.NegativeOne:
                return -pi/2
        if arg in [S.Infinity, S.NegativeInfinity, S.ComplexInfinity]:
            return S.Zero

        if arg.could_extract_minus_sign():
            return -cls(-arg)

        if arg.is_infinite:
            return S.Zero

        if arg.is_number:
            acsc_table = cls._acsc_table()
            if arg in acsc_table:
                return acsc_table[arg]

        if isinstance(arg, csc):
            ang = arg.args[0]
            if ang.is_comparable:
                ang %= 2*pi # restrict to [0,2*pi)
                if ang > pi: # restrict to (-pi,pi]
                    ang = pi - ang

                # restrict to [-pi/2,pi/2]
                if ang > pi/2:
                    ang = pi - ang
                if ang < -pi/2:
                    ang = -pi - ang

                return ang

        if isinstance(arg, sec): # asec(x) + acsc(x) = pi/2
            ang = arg.args[0]
            if ang.is_comparable:
                return pi/2 - asec(arg)

    def fdiff(self, argindex=1):
        if argindex == 1:
            return -1/(self.args[0]**2*sqrt(1 - 1/self.args[0]**2))
        else:
            raise ArgumentIndexError(self, argindex)

    def inverse(self, argindex=1):
        """
        Returns the inverse of this function.
        """
        return csc

    @staticmethod
    @cacheit
    def taylor_term(n, x, *previous_terms):
        if n == 0:
            return pi/2 - S.ImaginaryUnit*log(2) + S.ImaginaryUnit*log(x)
        elif n < 0 or n % 2 == 1:
            return S.Zero
        else:
            x = sympify(x)
            if len(previous_terms) > 2 and n > 2:
                p = previous_terms[-2]
                return p * ((n - 1)*(n-2)) * x**2/(4 * (n//2)**2)
            else:
                k = n // 2
                R = RisingFactorial(S.Half, k) *  n
                F = factorial(k) * n // 2 * n // 2
                return S.ImaginaryUnit * R / F * x**n / 4

    def _eval_as_leading_term(self, x, logx=None, cdir=0):  # acsc
        arg = self.args[0]
        x0 = arg.subs(x, 0).cancel()
        if x0 is S.NaN:
            return self.func(arg.as_leading_term(x))
        # Handling branch points
        if x0 in (-S.One, S.One, S.Zero):
            return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        if x0 is S.ComplexInfinity:
            return (1/arg).as_leading_term(x)
        # Handling points lying on branch cuts (-1, 1)
        if x0.is_real and (1 - x0**2).is_positive:
            ndir = arg.dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if x0.is_positive:
                    return pi - self.func(x0)
            elif im(ndir).is_positive:
                if x0.is_negative:
                    return -pi - self.func(x0)
            else:
                return self.rewrite(log)._eval_as_leading_term(x, logx=logx, cdir=cdir).expand()
        return self.func(x0)

    def _eval_nseries(self, x, n, logx, cdir=0):  # acsc
        from sympy.series.order import O
        arg0 = self.args[0].subs(x, 0)
        # Handling branch points
        if arg0 is S.One:
            t = Dummy('t', positive=True)
            ser = acsc(S.One + t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.NegativeOne + self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        if arg0 is S.NegativeOne:
            t = Dummy('t', positive=True)
            ser = acsc(S.NegativeOne - t**2).rewrite(log).nseries(t, 0, 2*n)
            arg1 = S.NegativeOne - self.args[0]
            f = arg1.as_leading_term(x)
            g = (arg1 - f)/ f
            res1 = sqrt(S.One + g)._eval_nseries(x, n=n, logx=logx)
            res = (res1.removeO()*sqrt(f)).expand()
            return ser.removeO().subs(t, res).expand().powsimp() + O(x**n, x)

        res = Function._eval_nseries(self, x, n=n, logx=logx)
        if arg0 is S.ComplexInfinity:
            return res
        # Handling points lying on branch cuts (-1, 1)
        if arg0.is_real and (1 - arg0**2).is_positive:
            ndir = self.args[0].dir(x, cdir if cdir else 1)
            if im(ndir).is_negative:
                if arg0.is_positive:
                    return pi - res
            elif im(ndir).is_positive:
                if arg0.is_negative:
                    return -pi - res
            else:
                return self.rewrite(log)._eval_nseries(x, n, logx=logx, cdir=cdir)
        return res

    def _eval_rewrite_as_log(self, arg, **kwargs):
        return -S.ImaginaryUnit*log(S.ImaginaryUnit/arg + sqrt(1 - 1/arg**2))

    _eval_rewrite_as_tractable = _eval_rewrite_as_log

    def _eval_rewrite_as_asin(self, arg, **kwargs):
        return asin(1/arg)

    def _eval_rewrite_as_acos(self, arg, **kwargs):
        return pi/2 - acos(1/arg)

    def _eval_rewrite_as_atan(self, x, **kwargs):
        return sqrt(x**2)/x*(pi/2 - atan(sqrt(x**2 - 1)))

    def _eval_rewrite_as_acot(self, arg, **kwargs):
        return sqrt(arg**2)/arg*(pi/2 - acot(1/sqrt(arg**2 - 1)))

    def _eval_rewrite_as_asec(self, arg, **kwargs):
        return pi/2 - asec(arg)


class atan2(InverseTrigonometricFunction):
    r"""
    The function ``atan2(y, x)`` computes `\operatorname{atan}(y/x)` taking
    two arguments `y` and `x`.  Signs of both `y` and `x` are considered to
    determine the appropriate quadrant of `\operatorname{atan}(y/x)`.
    The range is `(-\pi, \pi]`. The complete definition reads as follows:

    .. math::

        \operatorname{atan2}(y, x) =
        \begin{cases}
          \arctan\left(\frac y x\right) & \qquad x > 0 \\
          \arctan\left(\frac y x\right) + \pi& \qquad y \ge 0, x < 0 \\
          \arctan\left(\frac y x\right) - \pi& \qquad y < 0, x < 0 \\
          +\frac{\pi}{2} & \qquad y > 0, x = 0 \\
          -\frac{\pi}{2} & \qquad y < 0, x = 0 \\
          \text{undefined} & \qquad y = 0, x = 0
        \end{cases}

    Attention: Note the role reversal of both arguments. The `y`-coordinate
    is the first argument and the `x`-coordinate the second.

    If either `x` or `y` is complex:

    .. math::

        \operatorname{atan2}(y, x) =
            -i\log\left(\frac{x + iy}{\sqrt{x^2 + y^2}}\right)

    Examples
    ========

    Going counter-clock wise around the origin we find the
    following angles:

    >>> from sympy import atan2
    >>> atan2(0, 1)
    0
    >>> atan2(1, 1)
    pi/4
    >>> atan2(1, 0)
    pi/2
    >>> atan2(1, -1)
    3*pi/4
    >>> atan2(0, -1)
    pi
    >>> atan2(-1, -1)
    -3*pi/4
    >>> atan2(-1, 0)
    -pi/2
    >>> atan2(-1, 1)
    -pi/4

    which are all correct. Compare this to the results of the ordinary
    `\operatorname{atan}` function for the point `(x, y) = (-1, 1)`

    >>> from sympy import atan, S
    >>> atan(S(1)/-1)
    -pi/4
    >>> atan2(1, -1)
    3*pi/4

    where only the `\operatorname{atan2}` function reurns what we expect.
    We can differentiate the function with respect to both arguments:

    >>> from sympy import diff
    >>> from sympy.abc import x, y
    >>> diff(atan2(y, x), x)
    -y/(x**2 + y**2)

    >>> diff(atan2(y, x), y)
    x/(x**2 + y**2)

    We can express the `\operatorname{atan2}` function in terms of
    complex logarithms:

    >>> from sympy import log
    >>> atan2(y, x).rewrite(log)
    -I*log((x + I*y)/sqrt(x**2 + y**2))

    and in terms of `\operatorname(atan)`:

    >>> from sympy import atan
    >>> atan2(y, x).rewrite(atan)
    Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)), (pi, re(x) < 0), (0, Ne(x, 0)), (nan, True))

    but note that this form is undefined on the negative real axis.

    See Also
    ========

    sin, csc, cos, sec, tan, cot
    asin, acsc, acos, asec, atan, acot

    References
    ==========

    .. [1] https://en.wikipedia.org/wiki/Inverse_trigonometric_functions
    .. [2] https://en.wikipedia.org/wiki/Atan2
    .. [3] https://functions.wolfram.com/ElementaryFunctions/ArcTan2

    """

    @classmethod
    def eval(cls, y, x):
        from sympy.functions.special.delta_functions import Heaviside
        if x is S.NegativeInfinity:
            if y.is_zero:
                # Special case y = 0 because we define Heaviside(0) = 1/2
                return pi
            return 2*pi*(Heaviside(re(y))) - pi
        elif x is S.Infinity:
            return S.Zero
        elif x.is_imaginary and y.is_imaginary and x.is_number and y.is_number:
            x = im(x)
            y = im(y)

        if x.is_extended_real and y.is_extended_real:
            if x.is_positive:
                return atan(y/x)
            elif x.is_negative:
                if y.is_negative:
                    return atan(y/x) - pi
                elif y.is_nonnegative:
                    return atan(y/x) + pi
            elif x.is_zero:
                if y.is_positive:
                    return pi/2
                elif y.is_negative:
                    return -pi/2
                elif y.is_zero:
                    return S.NaN
        if y.is_zero:
            if x.is_extended_nonzero:
                return pi*(S.One - Heaviside(x))
            if x.is_number:
                return Piecewise((pi, re(x) < 0),
                                 (0, Ne(x, 0)),
                                 (S.NaN, True))
        if x.is_number and y.is_number:
            return -S.ImaginaryUnit*log(
                (x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))

    def _eval_rewrite_as_log(self, y, x, **kwargs):
        return -S.ImaginaryUnit*log((x + S.ImaginaryUnit*y)/sqrt(x**2 + y**2))

    def _eval_rewrite_as_atan(self, y, x, **kwargs):
        return Piecewise((2*atan(y/(x + sqrt(x**2 + y**2))), Ne(y, 0)),
                         (pi, re(x) < 0),
                         (0, Ne(x, 0)),
                         (S.NaN, True))

    def _eval_rewrite_as_arg(self, y, x, **kwargs):
        if x.is_extended_real and y.is_extended_real:
            return arg_f(x + y*S.ImaginaryUnit)
        n = x + S.ImaginaryUnit*y
        d = x**2 + y**2
        return arg_f(n/sqrt(d)) - S.ImaginaryUnit*log(abs(n)/sqrt(abs(d)))

    def _eval_is_extended_real(self):
        return self.args[0].is_extended_real and self.args[1].is_extended_real

    def _eval_conjugate(self):
        return self.func(self.args[0].conjugate(), self.args[1].conjugate())

    def fdiff(self, argindex):
        y, x = self.args
        if argindex == 1:
            # Diff wrt y
            return x/(x**2 + y**2)
        elif argindex == 2:
            # Diff wrt x
            return -y/(x**2 + y**2)
        else:
            raise ArgumentIndexError(self, argindex)

    def _eval_evalf(self, prec):
        y, x = self.args
        if x.is_extended_real and y.is_extended_real:
            return super()._eval_evalf(prec)