Prove that \[ tan^{-1} \bigg( \frac{\sqrt {1+x}-\sqrt {1-x}}{\sqrt {1+x}+\sqrt {1-x}} \bigg) = \frac{\pi}{4} - \frac{1}{2} cos^{-1}x , \frac{-1}{\sqrt 2} \leq x \: \leq 1 \]

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