Question

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

• A. $\frac{3\pi }{4}$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. none of these

Answer 1

Answer: (B)

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