Question

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

Answer 1

Given expression ${\sin }^{-1}\left[\cot [{\sin }^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right)+{\cos }^{-1}\left(\frac{\sqrt{12}}{4}\right)+{\sec }^{-1}\sqrt{2}]\right]$

$={\sin }^{-1}\left[\cot [{\sin }^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right)+{\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)+{\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)]\right]$

$={\sin }^{-1}\left[\cot ({\sin }^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right)+{\cos }^{-1}\left(\sqrt{\frac{2-\sqrt{3}}{4}}\right))\right]$

$={\sin }^{-1}\left(\cot \frac{\pi }{2}\right)={\sin }^{-1}0=0$