Question

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

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

Answer 1

Answer: (A)

$0$

Given

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

${\sin }^{-1}\sqrt{\frac{2-\sqrt{3}}{4}}$

$⟹{\sin }^{-1}\sqrt{\frac{4-2\sqrt{3}}{8}}$

$⟹{\sin }^{-1}\sqrt{\frac{(\sqrt{3}{)}^2+1^2-2(1\left)\right(\sqrt{3})}{8}}$

$⟹{\sin }^{-1}\sqrt{{\left(\frac{\left(\sqrt{3}\right)-\left(1\right)}{2\sqrt{2}}\right)}^2}$

$⟹{\sin }^{-1}\sqrt{\frac{\left(\sqrt{3}\right)-\left(1\right)}{2\sqrt{2}}}$

$⟹{15}^{\circ }$

${\cos }^{-1}\frac{\sqrt{12}}{4}$

$⟹{\cos }^{-1}\frac{2\sqrt{3}}{4}$

$⟹{\cos }^{-1}\frac{\sqrt{3}}{2}$

$⟹{30}^{\circ }$

${\sec }^{-1}\sqrt{2}$

$⟹{45}^{\circ }$

$⟹{\sin }^{-1}\left(\cot (15+30+45)\right)$

$⟹{\sin }^{-1}\left(\cot \left(90\right)\right)$

$⟹{\sin }^{-1}\left(0\right)$

$⟹0$