Question:
Mark the tick against the correct answer in the following:
$\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)=?$
A. $\frac{\pi}{3}$
B. $\frac{-\pi}{3}$
C. $\frac{5 \pi}{3}$
D. none of these
Solution:
To Find: The value of $\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)$
Let, $x=\tan ^{-1}(\sqrt{3})-\sec ^{-1}(-2)$
$\Rightarrow \mathrm{x}=\frac{\pi}{3}-\left[\pi-\sec ^{-1}(2)\right]\left(\because \tan \left(\frac{\pi}{3}\right)=\sqrt{3}\right.$ and $\left.\sec ^{-1}(-\theta)=\pi-\sec ^{-1}(\theta)\right)$
$\Rightarrow \mathrm{x}=\frac{\pi}{3}-\left[\pi-\frac{\pi}{3}\right]$
$\Rightarrow \mathrm{x}=\frac{\pi}{3}-\left[\frac{2 \pi}{3}\right]$
$\Rightarrow \mathrm{x}=-\frac{\pi}{3}$