# Show that

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

Solution:

(a) $x$

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

Then,

$\sin \left[\cot ^{-1}\left\{\tan \left(\cos ^{-1} x\right)\right\}\right]=\sin \left[\cot ^{-1}\{\tan y\}\right]$

$=\sin \left[\cot ^{-1}\left\{\cot \left(\frac{\pi}{2}-y\right)\right\}\right]$

$=\sin \left(\frac{\pi}{2}-y\right)$

$=\cos y$

$=x \quad[\because \cos y=x]$