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

• Correct answer
  A

  $\frac{3 \sqrt{7}}{8}$
• B

  $\frac{5 \sqrt{7}}{8}$
• C

  $\frac{5 \sqrt{7}}{2}$
• D

  $\frac{3 \sqrt{7}}{2}$

Solution

$=\sin \left[2 \cos ^{-1}\left\{\cot \left(\tan ^{-1} \frac{1}{1-\frac{1}{4}}\right)\right\}\right]$ $=\sin \left[2 \cos ^{-1}\left\{\cot \left(\tan ^{-1} \frac{\frac{4}{3}}{3}\right)\right\}\right]$ $=\sin \left[2 \cos ^{-1} \frac{3}{\frac{4}{4}}\right]=\sin 20$ $\left[\right.$ where, $\cos \theta=\frac{3}{4}$ and $\left.\theta=\cos ^{-1} \frac{3}{4}\right]$

$=2 \sin \theta \cos \theta=2 \frac{\sqrt{7}}{4}, \frac{3}{4}=\frac{3 \sqrt{7}}{8}$

Hence, the correct answer is option: 'A'.