Question

Rewrite each expression such that the argument x is positive. a. cot(−x)cos(−x) sin(−x)

Answer 1

$\cos(x)$ is an even function, while $\sin(x)$ is odd. This means

$\cos(-x) = \cos(x) \text{ and } \sin(-x) = -\sin(x)$

$\cot(x)$ is defined by

$\cot(x) = \dfrac{\cos(x)}{\sin(x)}$

so it is an odd function, since

$\cot(-x) = \dfrac{\cos(-x)}{\sin(-x)} = \dfrac{\cos(x)}{-\sin(x)} = -\cot(x)$

Putting everything together, it follows that

$\cot(-x) \cos(-x) \sin(-x) = (-\cot(x)) \cos(x) (-\sin(x)) \\\\~~~~~~~~= \cot(x) \cos(x) \sin(x) \\\\ ~~~~~~~~= \cos^2(x)$