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)

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