Question

If $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\text{cosec}x\right)$, then the value of $x$ is:

• A. $\frac{3\pi }{4}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is B $\frac{\pi }{4}$

We have $2{\tan }^{-1}\left(\cos x\right)={\tan }^{-1}\left(2\text{cosec}x\right)$
$\Rightarrow {\tan }^{-1}\frac{2\cos x}{1-{\cos }^{2}x}={\tan }^{-1}\left(2\text{cosec}x\right)\Rightarrow \frac{2\cos x}{{\sin }^{2}x}=2\text{cosec}x\Rightarrow 2\cos x=2\sin x\therefore x=\frac{\pi }{4}$