Q. The value of $\cos{\lbrack\tan^{- 1}{\{\sin{(\cot^{- 1}x)}\}}\rbrack}$ is
A) $\sqrt{\frac{x^{2} + 1}{x^{2} - 1}}$
B) $\sqrt{\frac{{1 - x}^{2}}{x^{2} + 2}}$
C) $\sqrt{\frac{{1 - x}^{2}}{{1 + x}^{2}}}$
D) $\sqrt{\frac{x^{2} + 1}{x^{2} + 2}}$
Explanation
(d)
$\cos{\lbrack\tan^{- 1}{\{\sin{(\cot^{- 1}x)}\}}\rbrack}$
$= \cos\left\lbrack \tan^{- 1}\left\{ \sin\left( \sin^{- 1}\frac{1}{\sqrt{1 + x^{2}}} \right) \right\} \right\rbrack$
$= \cos\left\lbrack \tan^{- 1}\frac{1}{\sqrt{1 + x^{2}}} \right\rbrack$
$= \cos\left\lbrack \cos^{- 1}\sqrt{\frac{1 + x^{2}}{2 + x^{2}}} \right\rbrack$
$= \sqrt{\frac{1 + x^{2}}{2 + x^{2}}}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.