sines, cosines, and hyperbolic functions as follows: By taking advantage of domain coloring, it is possible to graph the trigonometric functions as complex-valued functions. Various features unique to the complex functions can be seen from the graph; for example, the sine and cosine functions can be seen to be unbounded as the imaginary part of becomes larger (since the color white represents infinity), and the fact that the functions contain simple zeros or poles is apparent from the fact that the hue cycles around each zero or pole exactly once. Comparing these graphs with those of the corresponding Hyperbolic functions highlights the relationships between the two. Basic identities Many identities interrelate the trigonometric functions. This section contains the most basic ones; for more identities, see List of trigonometric identities. These identities may be proved geometrically from the unit-circle definitions or the right-angled-triangle definitions (although, for the latter definitions, care must be taken for angles that are not in the interval , see Proofs of trigonometric identities). For non-geometrical proofs using only tools of calculus, one may use directly the differential equations, in a way that is similar to that of the above proof of Euler's identity. One can also use Euler's identity for expressing all trigonometric functions in terms of complex exponentials and using properties of the exponential function. Parity The cosine and the secant are even functions; the other trigonometric functions are odd functions. That is: Periods All trigonometric functions are periodic functions of period . This is the smallest period, except for the tangent and the cotangent, which have as smallest period. This means that, for every integer , one has Pythagorean identity The Pythagorean identity, is the expression of the Pythagorean theorem in terms of trigonometric functions. It is . Dividing through by either or gives and . Sum and difference formulas The sum and difference formulas allow expanding the sine, the cosine, and the tangent of a sum or a difference of two angles in terms of sines and cosines and tangents of the angles themselves. These can be derived geometrically, using arguments that date to Ptolemy. One can also produce them algebraically using Euler's formula. Sum Difference When the two angles are equal, the sum formulas reduce to simpler equations known as the double-angle formulae. These identities can be used to derive the product-to-sum identities. By setting all trigonometric functions of can be expressed as