Question

The value of ${\sin }^{-1}\left\{\cot \{{\sin }^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}+{\cos }^{-1}\frac{\sqrt{12}}{4}+{\sec }^{-1}\sqrt{2}\}\right\}$ is

• A. $\frac{\pi }{4}$
• B. $\frac{\pi }{6}$
• C. $0$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (C)

$0$

We have, ${\sin }^{-1}\left\{\cot ({\sin }^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}+{\cos }^{-1}\frac{\sqrt{12}}{4}+{\sec }^{-1}\sqrt{2})\right\}$
$={\sin }^{-1}\left\{\cot ({\sin }^{-1}\frac{\sqrt{3}-1}{2\sqrt{2}}+{\cos }^{-1}\frac{\sqrt{3}}{2}+{\sec }^{-1}\sqrt{2})\right\}$
$={\sin }^{-1}\left\{\cot (\frac{\pi }{12}+\frac{\pi }{6}+\frac{\pi }{4})\right\}$
$={\sin }^{-1}\left(\cot \frac{\pi }{2}\right)$
$={\sin }^{-1}0=0$