Question

Solve $\cot \left(x/2\right)-\cos \left(x/2\right)=\cot x$

Answer 1

$\cot \left(\frac{x}{2}\right)-\csc \left(\frac{x}{2}\right)=\cot x$
Put $y=\tan \left(\frac{x}{4}\right)$
$\frac{y^2+1}{2y\left(y^{2-}1\right)}-\frac{y(y^2+1)}{2(y^2-1)}-\frac{(y^2+1)2}{4y(y^2-1)}=0$
$\Rightarrow -\frac{(y^2+1)({3y}^2-1)}{4y(y^2-1)}=0$
$\Rightarrow (y^2+1)$ or $({3y}^2-1)=0$
$\Rightarrow y^2\ne -1$ or $3y^2=1$
$\Rightarrow y^2=\frac{1}{3}$
$\Rightarrow \tan \left(\frac{x}{4}\right)=\frac{1}{\sqrt{3}}$ or $\tan \left(\frac{x}{4}\right)=\frac{-1}{\sqrt{3}}$
$\Rightarrow x=4n\pi +\frac{2\pi }{3}$ or $x=4n\pi -\frac{2\pi }{3}$
$\Rightarrow x=4n\pi \pm \frac{2\pi }{3}$