A) \[\frac{\sqrt{1-{{x}^{2}}}}{x}\]
B) \[\frac{x}{\sqrt{1-{{x}^{2}}}}\]
C) \[x\]
D) None of these
Correct Answer: C
Solution :
\[\sin [{{\cot }^{-1}}\{\tan ({{\cos }^{-1}}x)\}]\] \[=\sin \left[ {{\cot }^{-1}}\left\{ \tan \left( {{\tan }^{-1}}\frac{\sqrt{1-{{x}^{2}}}}{x} \right) \right\} \right]\] \[\left[ \because \,\,{{\cos }^{-1}}x={{\tan }^{-1}}\left( \frac{\sqrt{1-{{x}^{2}}}}{x} \right) \right]\] \[=\sin \left[ {{\cot }^{-1}}\frac{\sqrt{1-{{x}^{2}}}}{x} \right]\] \[=\sin [{{\sin }^{-1}}x]\]\[\left[ \because \,\,{{\sin }^{-1}}x={{\cot }^{-1}}\frac{\sqrt{1-{{x}^{2}}}}{x} \right]\] \[=x\]
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