Question

Verify the identity. (1 point) cot x minus pi divided by two. = -tan x

Answer 1

Apply the definitions:

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

which implies

$\cot\left(x-\dfrac{\pi}{2}\right) = \dfrac{\cos\left(x-\dfrac{\pi}{2}\right)}{\sin\left(x-\dfrac{\pi}{2}\right)}$

We also have the identities

$\cos\left(x-\dfrac{\pi}{2}\right) = -\sin(x)$

$\sin\left(x-\dfrac{\pi}{2}\right) = \cos(x)$

Which means

$\cot\left(x-\dfrac{\pi}{2}\right) = \dfrac{\cos\left(x-\dfrac{\pi}{2}\right)}{\sin\left(x-\dfrac{\pi}{2}\right)} = \dfrac{-\sin(x)}{\cos(x)} = -\tan(x)$