91Ó°ÊÓ

For any angle \(x\) in the unit circle, the cosine of \(x\) is the x-coordinate of the point where the terminal side of \(x\) intersects the unit circle. Thus, \(\cos(-x)\) would be the x-coordinate of the point where the terminal side of \(-x\) intersects the unit circle. But because of the symmetry of the circle, \(\cos(-x) = \cos(x)\). Thus, cosine is an even function. Secant is the reciprocal of cosine. Therefore, \(\sec(-x) = 1/\cos(-x) = 1/\cos(x) = \sec(x)\). Thus, secant is also an even function.

Similar to cosine, for any angle \(x\) in the unit circle, the sine of \(x\) is the y-coordinate of the point where the terminal side of \(x\) intersects the unit circle. Thus, \(\sin(-x)\) would be the y-coordinate of the point where the terminal side of \(-x\) intersects the unit circle. But because of the symmetry of the circle, \(\sin(-x) = -\sin(x)\). Thus, sine is an odd function. Cosecant is the reciprocal of sine. Therefore, \(\csc(-x) = 1/\sin(-x) = -1/\sin(x) = -\csc(x)\). Thus, cosecant is also an odd function.Tangent is sine over cosine, and cotangent is the reciprocal of tangent. Thus, \(\tan(-x) = \sin(-x)/\cos(-x) = -\sin(x)/\cos(x) = -\tan(x)\) and \(\cot(-x) = 1/\tan(-x) = -1/\tan(x) = -\cot(x)\). Thus, tangent and cotangent are also odd functions.