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