Question

Determine whether the given function is even or odd $\frac{\cos x\sin x}{\tan x+\cot x}$

Answer 1

$f\left(x\right)=\frac{\cos x\sin x}{\tan x+\cot x}$

$f(-x)=\frac{\cos (-x)\sin (-x)}{\tan (-x)+\cot (-x)}$

$=\frac{-\cos x\sin x}{-(\tan x+\cot x)}$ (Since, sin , tan , cot are all odd functions while cos is an even function)

$=\frac{\cos x\sin x}{\tan x+\cot x}$

$\Rightarrow f(-x)=f\left(x\right)$
Therefore, $f$ is even.