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  4. sin [2 cos-1 cot (2 tan-1 (1/2)) ] is equal to

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Q. $\sin \left[2 \cos^{-1} \cot \left(2 \tan^{-1} \frac{1}{2}\right) \right] $ is equal to

COMEDKCOMEDK 2012Inverse Trigonometric Functions

Solution:

$\sin \left[2 \cos^{-1} \cot \left(2 \tan^{-1} \frac{1}{2}\right) \right] $
$ = \sin \left[2 \cos ^{-1} \cot \left\{ \tan^{-1} \left(\frac{2\left(\frac{1}{2}\right)}{1- \frac{1}{4}}\right)\right\} \right]$
$ = \sin \left[2 \cos ^{-1} \cot \left\{ \tan ^{-1}\left(\frac{4}{3}\right) \right\}\right] $
$ = \sin \left[2 \cos ^{-1} \cot \left( \cot^{-1} \left(\frac{3}{4}\right)\right) \right] $
$ = \sin \left[2 \cos ^{-1} \left(\frac{3}{4}\right) \right] = \sin \left[ \cos ^{-1} \left(2 \left(\frac{9}{16}\right) -1\right) \right] $
$ = \sin \left[ \cos ^{-1}\left(\frac{9}{8} -1\right) \right] = \sin \left[ \cos ^{-1} \left(\frac{1}{8}\right) \right] $
$ =\sin \left[\sin^{-1} \left(\sqrt{1- \frac{1}{64}}\right) \right] $
$ = \sqrt{\frac{63}{64}} = \frac{\sqrt{9 \times7}}{8}= \frac{3 \sqrt{ 7}}{8}$