Question:
Write the principal value of $\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$
Solution:
We have,
$\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$
$=\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left\{\sin \left(\pi-\frac{\pi}{3}\right)\right\}$ $\left[\because\left(\pi-\frac{2 \pi}{3}\right) \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$
$=\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left\{\sin \left(\frac{\pi}{3}\right)\right\}$
$=\frac{2 \pi}{3}+\frac{\pi}{3}$
$=\pi$
$\therefore \cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\pi$