Find the value of: $\;$ $\cot^{-1} \left\{\dfrac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}\right\}$, $\;\;$ $0 < x < \dfrac{\pi}{2}$
$\cot^{-1} \left\{\dfrac{\sqrt{1 - \sin x} + \sqrt{1 + \sin x}}{\sqrt{1 - \sin x} - \sqrt{1 + \sin x}}\right\}$
$= \cot^{-1} \left\{\dfrac{\left(\sqrt{1 - \sin x} + \sqrt{1 + \sin x}\right)^2}{\left(\sqrt{1 - \sin x} - \sqrt{1 + \sin x}\right) \left(\sqrt{1 - \sin x} + \sqrt{1 + \sin x}\right)} \right\}$
$= \cot^{-1} \left\{\dfrac{1 - \sin x + 1 + \sin x + 2 \sqrt{\left(1 - \sin x\right) \left(1 + \sin x\right)}}{1 - \sin x - 1 - \sin x} \right\}$
$= \cot^{-1} \left\{\dfrac{2 + 2 \sqrt{1 - \sin^2 x}}{- 2 \sin x} \right\}$
$= \cot^{-1} \left\{\dfrac{2 + 2 \cos x}{- 2 \sin x} \right\}$
$= \cot^{-1} \left\{\dfrac{1 + \cos x}{- \sin x} \right\}$
$= \cot^{-1} \left\{\dfrac{2 \cos^2 \left(\dfrac{x}{2}\right)}{- 2 \sin \left(\dfrac{x}{2}\right) \cos \left(\dfrac{x}{2}\right)} \right\}$
$= \cot^{-1} \left\{- \cot \left(\dfrac{x}{2}\right) \right\}$
$= \pi - \cot^{-1} \left\{\cot \left(\dfrac{x}{2}\right) \right\}$
$= \pi - \dfrac{x}{2}$