Find the principal value of $\tan^{-1}(1) + \cos^{-1}\left( -\frac{1}{2} \right) + \sin^{-1}\left( -\frac{1}{\sqrt{2}} \right)$.

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

The correct answer is Option (2) → $\frac{2\pi}{3}$ ##

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

$= \tan^{-1} \left( \tan \frac{\pi}{4} \right) + \cos^{-1} \left( \cos \frac{2\pi}{3} \right) + \sin^{-1} \left( \sin \left( -\frac{\pi}{4} \right) \right)$

$= \frac{\pi}{4} + \frac{2\pi}{3} - \frac{\pi}{4} = \frac{2\pi}{3}$