Find the value of: $\;$ $\sin^{-1} \left\{ \cos \left[\sin^{-1} \left(x\right)\right]\right\} + \cos^{-1} \left\{\sin \left[\cos^{-1} \left(x\right)\right] \right\}$
$\left[\text{Note: } \sin^{-1} \left(x\right) = \cos^{-1} \left(\sqrt{1 - x^2}\right), \;\;\; 0 < x < 1 \right.$
$\left. \cos^{-1} \left(x\right) = \sin^{-1} \left(\sqrt{1 - x^2}\right), \;\;\; 0 < x < 1 \right.$
$\left. \sin^{-1} \left(x\right) + \cos^{-1} \left(x\right) = \dfrac{\pi}{2}, \;\;\; \left|x\right| \leq 1 \right]$
$\sin^{-1} \left\{ \cos \left[\sin^{-1} \left(x\right)\right]\right\} + \cos^{-1} \left\{\sin \left[\cos^{-1} \left(x\right)\right] \right\}$
$= \sin^{-1} \left\{\cos \left[\cos^{-1} \left(\sqrt{1 - x^2}\right)\right] \right\} + \cos^{-1} \left\{\sin \left[\sin^{-1} \left(\sqrt{1 - x^2}\right)\right] \right\}$
$= \sin^{-1} \left\{\sqrt{1 - x^2} \right\} + \cos^{-1} \left\{\sqrt{1 - x^2} \right\}$
$= \dfrac{\pi}{2}$