Question

The value of $\tan ^{-1}\left(\frac{-1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left(\sin \left(-\frac{\pi}{2}\right)\right)$ is

• A. $\frac{\pi}{6}$
• B. $\frac{\pi}{12}$
• C. $-\frac{\pi}{12}$
• D. $\frac{\pi}{3}$

Answer 1

Answer: (C)

Given, $\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left[\sin \left(\frac{-\pi}{2}\right)\right]$
$=\tan ^{-1}\left(-\frac{1}{\sqrt{3}}\right)+\cot ^{-1}\left(\frac{1}{\sqrt{3}}\right)+\tan ^{-1}\left(-\sin \frac{\pi}{2}\right) [\because \sin (-\theta)=-\sin \theta]$
$=\frac{-\pi}{6}+\frac{\pi}{3}-\tan ^{-1}\left(\sin \frac{\pi}{2}\right)[\because \tan^{-1} (-x) = -\tan^{-1} x]$
$=\frac{\pi}{6}-\tan ^{-1}(1)=\frac{\pi}{6}-\frac{\pi}{4}=-\frac{\pi}{12}$