Write the principal value

We have,

$\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)$

$=\cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left\{\sin \left(\pi-\frac{\pi}{3}\right)\right\}$                  $\left[\because\left(\pi-\frac{2 \pi}{3}\right) \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$

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

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

$=\pi$

$\therefore \cos ^{-1}\left(\cos \frac{2 \pi}{3}\right)+\sin ^{-1}\left(\sin \frac{2 \pi}{3}\right)=\pi$