Question

The value of $\tan ^{-1}\left[2 \sin \left(2 \cos ^{-1} \frac{\sqrt{3}}{2}\right)\right]$ is

• A. $\frac{\pi}{3}$
• B. $\frac{2 \pi}{3}$
• C. $\frac{-\pi}{3}$
• D. $\frac{\pi}{6}$

Answer 1

Answer: (A)

Given, $ \tan ^{-1} {\left[2 \sin \left(2 \cos ^{-1}\left(\frac{\sqrt{3}}{2}\right)\right)\right] }$
$ =\tan ^{-1}\left[2 \sin \left(2 \times \frac{\pi}{6}\right)\right]$
$ =\tan ^{-1}\left[2 \sin \left(\frac{\pi}{3}\right)\right] $
$ =\tan ^{-1}\left[2 \times \frac{\sqrt{3}}{2}\right]$
$ =\tan ^{-1} \sqrt{3}=\frac{\pi}{3}$