Question

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

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

Answer 1

The correct option is A $0$

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

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

$={\sin }^{-1}\left[\cot ({15}^0+{30}^0+{45}^0)\right]={\sin }^{-1}\left[\cot {90}^0\right]={\sin }^{-1}0=0$