Question

If $x=2{\cos }^{-1}\left[\frac{1}{2}\right]+{\sin }^{-1}\left[\frac{1}{\sqrt{2}}\right]+{\tan }^{-1}\left(\sqrt{3}\right)$ and $y=\cos \left[\frac{1}{2}{\sin }^{-1}\left[\sin \frac{x}{2}\right]\right]$ then which of the following statements holds good ?

• A. $y=\cos \frac{3\pi }{16}$
• B. $y=\cos \frac{5\pi }{16}$
• C. $x=4{\cos }^{-1}y$
• D. none of these

Answer 1

The correct option is A $y=\cos \frac{3\pi }{16}$

$x=\frac{2\pi }{3}+\frac{\pi }{4}+\frac{\pi }{3}=\frac{5\pi }{4}$

$y=\cos \left[\frac{1}{2}{\sin }^{-1}\left[\sin \frac{5\pi }{8}\right]\right]$

$=\cos \left[\frac{1}{2}[\pi -\frac{5\pi }{8}]\right]=\cos \frac{3\pi }{16}$.