Question

The value of $\cos \left[\frac{1}{2}{\cos }^{-1}\left[\cos [-\frac{14\pi }{5}]\right]\right]$is/are -

• A. $\cos [-\frac{7\pi }{5}]$
• B. $\sin \left[\frac{\pi }{10}\right]$
• C. $\cos \left[\frac{2\pi }{5}\right]$
• D. $-\cos \left[\frac{3\pi }{5}\right]$

Answer 1

Answer: (B), (C), (D)

$\sin \left[\frac{\pi }{10}\right]$
$\cos \left[\frac{2\pi }{5}\right]$
$-\cos \left[\frac{3\pi }{5}\right]$

Given, $\cos \left[\frac{1}{2}{\cos }^{-1}\cos [2\pi +\frac{4\pi }{5}]\right]$
$=\cos [\frac{1}{2}\times \frac{4\pi }{5}]=\cos \frac{2\pi }{5}$
$=\cos [\pi -\frac{3\pi }{5}]=-\cos \frac{3\pi }{5}$
& $\cos \frac{2\pi }{5}=\sin [\frac{\pi }{2}-\frac{2\pi }{5}]=\sin \left[\frac{\pi }{10}\right]$