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

$\tan x \cot \frac{x}{2}+\tan \frac{x}{2} \cot x=2$
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.