Question

If $x=2{\cos }^{-1}\left(\frac{1}{2}\right)+{\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+{\tan }^{-1}(\sqrt{3})$ and $y=\cos \left(\frac{1}{2}{\sin }^{-1}\left(\sin \frac{x}{2}\right)\right)$ then which one 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

Answer: (A)

$x=2{\cos }^{-1}\left(\frac{1}{2}\right)+{\sin }^{-1}\left(\frac{1}{\sqrt{2}}\right)+{\tan }^{-1}\sqrt{3}=\frac{2\pi }{3}+\frac{\pi }{4}+\frac{\pi }{3}=\frac{5\pi }{4}$
$y=\cos \left(\frac{1}{2}{\sin }^{-1}\left(\sin \frac{5\pi }{8}\right)\right)=\cos \left(\frac{1}{2}{\sin }^{-1}\sin \frac{3\pi }{8}\right)$
$y=\cos \left(\frac{1}{2}\cdot \frac{3\pi }{8}\right)=\cos \left(\frac{3\pi }{16}\right)$