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Question
Write the value of sin (cot−1 x).
Solution
We know
\[\cot^{- 1} x = \tan^{- 1} \frac{1}{x}\]
Now, we have
\[\sin\left( \cot^{- 1} x \right) = \sin\left( \tan^{- 1} \frac{1}{x} \right)\]
\[ = \sin\left[ \sin^{- 1} \left( \frac{\frac{1}{x}}{\sqrt{1 + \frac{1}{x^2}}} \right) \right] \left[ \because \tan^{- 1} x = \sin^{- 1} \left( \frac{x}{\sqrt{1 + x^2}} \right) \right]\]
\[ = \sin\left[ \sin^{- 1} \left( \frac{\frac{1}{x}}{\frac{\sqrt{x^2 + 1}}{x}} \right) \right]\]
\[ = \sin\left( \sin^{- 1} \frac{1}{\sqrt{x^2 + 1}} \right)\]
\[ = \frac{1}{\sqrt{x^2 + 1}} \left[ \because \sin\left( \sin^{- 1} x = x \right) \right]\]
Hence,
\[\sin\left( \cot^{- 1} x \right) = \frac{1}{\sqrt{x^2 - 1}}\]
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