Question

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

• A. $\frac{3\sqrt{7}}{8}$
• B. $\frac{5\sqrt{7}}{8}$
• C. $\frac{5\sqrt{7}}{2}$
• D. $\frac{3\sqrt{7}}{2}$

Answer 1

The correct option is A $\frac{3\sqrt{7}}{8}$

$\sin [2{\cos }^{-1}\cot \left({\tan }^{-1}\frac{1}{1-\frac{1}{4}}\right)=\sin \left[2{\cos }^{-1}\cot \left({\tan }^{-1}\frac{4}{3}\right)\right]$
$=\sin \left[2{\cos }^{-1}\cot \left({\cot }^{-1}\frac{3}{4}\right)\right]=\sin \left[2{\cos }^{-1}\frac{3}{4}\right]$
$=\sin 2\theta$ where $\cos \theta =\frac{3}{4}$
$=2\frac{\sqrt{7}}{4}\cdot \frac{3}{4}=\frac{3\sqrt{7}}{8}$