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

   

Answer 1

(a) x
Let \[\cos^{- 1} x = y\]
Then, 

\[\sin\left[ \cot^{- 1} \left\{ \tan\left( \cos^{- 1} x \right) \right\} \right] = \sin\left[ \cot^{- 1} \left\{ \tan y \right\} \right]\]
\[ = \sin\left[ \cot^{- 1} \left\{ \cot \left( \frac{\pi}{2} - y \right) \right\} \right] \]
\[ = \sin\left( \frac{\pi}{2} - y \right)\]
\[ = \cos{y} \]
\[ = x \left[ \because \cos{y} = x \right]\]