The value of $sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) - 2 sec^{-1}\left(2 tan\frac{\pi}{6}\right), $ is

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

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

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