Question

Number of values of $x\in (-\pi ,3\pi )-\left\{\frac{-\pi }{2},0,\frac{\pi }{2},\pi ,\frac{3\pi }{2},2\pi ,\frac{5\pi }{2}\right\}$ satisfying the equation $\tan x\cot \frac{x}{2}=2-\cot x\tan \frac{x}{2}$ is/are

Answer 1

Answer: 0

$\tan x\cot \frac{x}{2}+\tan \frac{x}{2}\cot x=2$
Let $\tan x\cot \frac{x}{2}=t$
$\Rightarrow t+\frac{1}{t}=2$
$\Rightarrow (t-1{)}^2=0$
$\Rightarrow t=1$
$\Rightarrow \tan x\cot \frac{x}{x}=1$
$\Rightarrow \sin x\cos \frac{x}{2}=\cos x\sin \frac{x}{2}$
$\Rightarrow \sin \left(\frac{x}{2}\right)=0$
$\Rightarrow \frac{x}{2}=0,\pi$ as $\frac{x}{2}\in \left(\frac{-\pi }{2},\frac{3\pi }{2}\right)$
$\Rightarrow x=0,2\pi$ both are not possible.