Calculate: $\;$ $\sin^{-1} \left(\dfrac{-\sqrt{2}}{2}\right) + \cos^{-1} \left(\dfrac{-1}{2}\right) - \tan^{-1} \left(- \sqrt{3}\right) + \cot^{-1} \left(\dfrac{-1}{\sqrt{3}}\right)$
Given: $\;$ $\sin^{-1} \left(\dfrac{-\sqrt{2}}{2}\right) + \cos^{-1} \left(\dfrac{-1}{2}\right) - \tan^{-1} \left(- \sqrt{3}\right) + \cot^{-1} \left(\dfrac{-1}{\sqrt{3}}\right)$
$= - \sin^{-1} \left(\dfrac{\sqrt{2}}{2}\right) + \pi - \cos^{-1} \left(\dfrac{1}{2}\right) + \tan^{-1} \left(\sqrt{3}\right) + \pi - \cot^{-1} \left(\dfrac{1}{\sqrt{3}}\right)$
$= - \sin^{-1} \left(\dfrac{1}{\sqrt{2}}\right) + 2 \pi - \dfrac{\pi}{3} + \dfrac{\pi}{3} - \dfrac{\pi}{3}$
$= \dfrac{- \pi}{4} + \dfrac{5 \pi}{3} = \dfrac{17 \pi}{12}$
$\left\{\because \; \sin^{-1} \left(-x\right) = - \sin^{-1} x, \; \forall \; x \in \left[-1, 1\right]\right\}$
$\left\{\cos^{-1} \left(-x\right) = \pi - \cos^{-1} x, \; \forall \; x \in \left[-1, 1\right]\right\}$
$\left\{\tan^{-1} \left(-x\right) = - \tan^{-1} x, \; \forall \; x \in R\right\}$
$\left\{\cot^{-1} \left(-x\right) = \pi - \cot^{-1} x, \; \forall \; x \in R\right\}$