Prove that \( \cos \left[\tan ^{-1}\left\{\sin \left(\cot ^{-1} x\r...

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Prove that \( \cos \left[\tan ^{-1}\left\{\sin \left(\cot ^{-1} x\right)\right\}=\sqrt{\frac{1+x^{2}}{2+x^{2}}}\right. \).
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