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Question
sin\[\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right]\] is equal to
Options
x
`sqrt(1-x^2`
`1/x`
none of these
Solution
(a) x
Let \[\cos^{- 1} x = y\]
Then,
\[\sin\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right] = \sin\left[ \cot^{- 1} \left\{ \tan y \right\} \right]\]
\[ = \sin\left[ \cot^{- 1} \left\{ \cot \left( \frac{\pi}{2} - y \right) \right\} \right] \]
\[ = \sin\left( \frac{\pi}{2} - y \right)\]
\[ = \cos{y} \]
\[ = x \left[ \because \cos{y} = x \right]\]
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