Question

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

Answer 1

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

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

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

$(\because \tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta })$

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

$cosecx\cot x=cosecx$

$cosecx=0$ or $\cot x=1$

$x=\varphi$ $x=n\pi +\frac{\pi }{4}$

$x=n\pi +\frac{\pi }{4}$.