= π/2 − θ) has cotangent equal to the length of the other leg, and cosecant equal to the length of the hypotenuse. In that way, this trigonometric identity involving the cotangent and the cosecant also follows from the Pythagorean theorem. The following table gives the identities with the factor or divisor that relates them to the main identity. Proof using the unit circle The unit circle centered at the origin in the Euclidean plane is defined by the equation: Given an angle θ, there is a unique point P on the unit circle at an anticlockwise angle of θ from the x-axis, and the x- and y-coordinates of P are: Consequently, from the equation for the unit circle: the Pythagorean identity. In the figure, the point P has a negative x-coordinate, and is appropriately given by x = cos θ, which is a negative number: cos θ = −cos(π−θ). Point P has a positive y-coordinate, and sin θ = sin(π−θ) > 0. As θ increases from zero to the full circle θ = 2π, the sine and cosine change signs in the various quadrants to keep x and y with the correct signs. The figure shows how the sign of the sine function varies as the angle changes quadrant. Because the x- and y-axes are perpendicular, this Pythagorean identity is equivalent to the Pythagorean theorem for triangles with hypotenuse of length 1 (which is in turn equivalent to the full Pythagorean theorem by applying a similar-triangles argument). See Unit circle for a short explanation. Proof using power series The trigonometric functions may also be defined using power series, namely (for x an angle measured in radians): Using the multiplication formula for power series at Multiplication and division of power series (suitably modified to account for the form of the series here) we obtain In the expression for sin2, n must be at least 1, while in the expression for cos2, the constant term is equal to 1. The remaining terms of their sum are (with common factors removed) by the binomial theorem. Consequently, which is the Pythagorean trigonometric identity. When the trigonometric functions are defined in this way, the identity in combination with the Pythagorean theorem shows that these power series parameterize the unit circle, which we used in the previous section. This definition constructs the sine and cosine functions in a rigorous fashion and proves that they are