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