# Is the function f(x) = 2 cot x even, odd or neither?

Nov 14, 2015

$2 \cot \left(x\right)$ is an odd function.

#### Explanation:

A function $f \left(x\right)$ is even if and only if $f \left(- x\right) = f \left(x\right)$
A function $f \left(x\right)$ is odd if and only if $f \left(- x\right) = - f \left(x\right)$

Note that $\sin \left(x\right)$ is an odd function and $\cos \left(x\right)$ is even.

Thus we have
$f \left(- x\right) = 2 \cot \left(- x\right) = 2 \cos \frac{- x}{\sin} \left(- x\right)$

Because sine is an odd function and cosine is even, we have

$f \left(- x\right) = 2 \cos \frac{x}{- \sin \left(x\right)} = - \left(2 \cos \frac{x}{\sin} \left(x\right)\right) = - f \left(x\right)$

Thus $2 \cot \left(x\right)$ is odd.