The principal values of $cos^{-1}\left(-sin\frac{7\pi}{6}\right)$, is

$\frac{5\pi}{3}$ $\frac{7\pi}{3}$ $\frac{\pi}{3}$ none of these

$\frac{\pi}{3}$

We have $cos^{-1}\left(-sin\frac{7\pi}{6}\right)= cos^{-1}\left(-sin\left(\pi +\frac{\pi}{6}\right)\right) = cos^{-1}\left(sin\frac{\pi}{6}\right)$ $=\frac{\pi}{2}-sin^{-1}\left(sin\frac{\pi}{6}\right)=\frac{\pi}{2}-\frac{\pi}{6}=\frac{\pi}{3}$