Question

sin $\left[{\cot }^{-1}\left\{\tan \left({\cos }^{-1}x\right)\right\}\right]$ is equal to
(a) x

(b) $\sqrt{1-x^2}$

(c) $\frac{1}{x}$

(d) none of these

Answer 1

(a) x

Let ${\cos }^{-1}x=y$

Then,
$$\begin{gathered} \sin \left[{\cot }^{-1}\left\{\tan \left({\cos }^{-1}x\right)\right\}\right]=\sin \left[{\cot }^{-1}\left\{\tan y\right\}\right] \\ =\sin \left[{\cot }^{-1}\left\{\cot \left(\frac{\mathrm{\pi }}{2}-y\right)\right\}\right] \\ =\sin \left(\frac{\mathrm{\pi }}{2}-y\right) \\ =\cos y \\ =x\left[\because \cos y=x\right] \end{gathered}$$