Question

The principal value of ${\tan }^{-1}\left[2\cos \left[2{\sin }^{-1}\frac{1}{2}\right]\right]$ is

• A. $\frac{\pi }{3}$
• B. $\frac{\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

The correct option is C $\frac{\pi }{4}$

${\tan }^{-1}\left[2\cos \left[2{\sin }^{-1}\frac{1}{2}\right]\right]$
$={\tan }^{-1}\left[2\cos [2\cdot \frac{\pi }{6}]\right]$
$={\tan }^{-1}\left[2\cos \frac{\pi }{3}\right]$
$={\tan }^{-1}[2\cdot \frac{1}{2}]$
$={\tan }^{-1}1$
$=\frac{\pi }{4}$