Prove that : $\cot^{-1} \bigg[\large \frac{\sqrt{1+\sin\: x}+\sqrt{1-\sin x}}{\sqrt{1+\sin x}-\sqrt{1-\sin x}} \bigg] = \large \frac{x}{2};$$x \in \bigg( 0, \large\frac{\pi}{2} \bigg).$

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