Question

Solve: $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2cosecx\right)$

Answer 1

$2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2co\sec x\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{2\cos x}{1-{\cos }^{2}x}\right)={\tan }^{-1}\left(2co\sec x\right)$

$\Rightarrow \frac{2\cos x}{1-{\cos }^{2}x}=2cosecx$

$\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=\frac{2}{\sin x}$

$\Rightarrow \cos x=\sin x$

$\Rightarrow \tan x=1$

$\therefore x=\frac{\pi }{4}$