Question

The value of ${\sin }^{-1}$$\left\{\tan ({\cos }^{-1}\sqrt{\frac{2+\sqrt{3}}{4}}+{\cos }^{-1}\frac{\sqrt{12}}{4}-{\text{cosec}}^{-1}\sqrt{2})\right\}$, is

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

Answer 1

Answer: (A)

$0$

Given:

$E={\sin }^{-1}\left(\tan ({\cos }^{-1}\frac{\sqrt{2+\sqrt{3}}}{2}+{\cos }^{-1}\frac{\sqrt{3}}{2}-{\text{cosec}}^{-1}\sqrt{2})\right)$

We know that,

$\cos {15}^0=\sqrt{\frac{2+\sqrt{3}}{4}}$

Therefore,
$\Rightarrow E={\sin }^{-1}\left(\tan \right({15}^0+{30}^0-{45}^0\left)\right)$
$\Rightarrow E={\sin }^{-1}\left(\tan 0^0\right)$

$\Rightarrow E={\sin }^{-1}0=0$