^ - 1 (sin(2π)/(3)) + ^ - 1 (cos(7π)/(6)) + ^ - 1 (tan(3π)/(4)) is...

Given, $\sin^{- 1}\left(sin\dfrac{2\pi}{3}\right) + \cos^{- 1}\left(cos\dfrac{7\pi}{6}\right) + \tan^{- 1}\tan \left(\dfrac{3\pi}{4}\right)$

Now simplifying each term we get,

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

$\cos^{- 1}\left(cos\dfrac{7\pi}{6}\right) = 2\pi - \dfrac{7\pi}{6} = \dfrac{5\pi}{6}$

$\tan^{- 1}\tan \left(\dfrac{3\pi}{4}\right) = \dfrac{3\pi}{4} - \pi = \dfrac{- \pi}{4}$

$\sin^{- 1}\left(sin\dfrac{2\pi}{3}\right) + \cos^{- 1}cos\dfrac{7\pi}{6} + \tan^{- 1}tan\dfrac{3\pi}{4}$

$= \dfrac{\pi}{3} + \dfrac{5\pi}{6} - \dfrac{\pi}{4}$

$= \dfrac{11\pi}{12}$