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}{4} \bigg).$

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