Question

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

Answer 1

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

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

$⟹\frac{2}{\sin x}=\frac{2\cos x}{1-{\cos }^{2}x}$

$2(1-{\cos }^{2}x)=2\sin x\cos x$

$⟹2-1-\cos 2x=\sin 2x$

$⟹\sin 2x+\cos 2x=1⟹\sin (2x+\frac{\pi }{4})=\frac{1}{\sqrt{2}}⟹2x+\frac{\pi }{4}=n\pi +(-1{)}^n\frac{\pi }{4}$

$x=\frac{n\pi }{2}+((-1{)}^n-1)\frac{\pi }{8}$