For the principal values, evaluate the following:
Question:

For the principal values, evaluate the following:

(i) $\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)$

(ii) $\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-2 \sec ^{-1}\left(2 \tan \frac{\pi}{6}\right)$

Solution:

(i)

$\tan ^{-1} \sqrt{3}-\sec ^{-1}(-2)=\tan ^{-1}\left(\tan \frac{\pi}{3}\right)-\sec ^{-1}\left(\sec \frac{2 \pi}{3}\right)$

$=\frac{\pi}{3}-\frac{2 \pi}{3}$

$=-\frac{\pi}{3}$

(ii)

$\sin ^{-1}\left(-\frac{\sqrt{3}}{2}\right)-2 \sec ^{-1}\left(2 \tan \frac{\pi}{6}\right)=-\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)-2 \sec ^{-1}\left(2 \times \frac{1}{\sqrt{3}}\right)$

$=-\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)-2 \sec ^{-1}\left(\frac{2}{\sqrt{3}}\right)$

$=-\sin ^{-1}\left(\sin \frac{\pi}{3}\right)-2 \sec ^{-1}\left(\sec \frac{\pi}{6}\right)$

$=-\frac{\pi}{3}-\frac{\pi}{3}$

$=-\frac{2 \pi}{3}$