Evaluate: `sin(cot^(-1)x)` (ii) `cos(tan^(-1)x)`

Explanation

Let `cot ^{-1} x=y Rightarrow x=cot y Rightarrow sin y=frac{1}{sqrt{1+x^{2}}}`
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tan ^{-1} frac{1}{sqrt{x^{2}+1}}=z Rightarrow tan z=frac{1}{sqrt{1+x^{2}}} `
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cos z=frac{1}{sqrt{1+(frac{1}{sqrt{x}+1})^{2}}}=frac{1}{sqrt{1+frac{1}{x+1}}}=frac{1}{sqrt{frac{x+2}{x+1}}}=sqrt{frac{x^{2}+1}{x^{2}+2}}`
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So cos [tan ^{-1}{sin (cot ^{-1} x)}]=cos [tan ^{-1}{sin y}]=cos [tan ^{-1} frac{1}{sqrt{1+x}}]=cos z=sqrt{frac{x^{2}+2}{x^{2}+1}}`
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