Question

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

• A. $\frac{\sqrt{3}}{2}$
• B. $-\frac{\sqrt{3}}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Answer 1

The correct option is C

$\frac{1}{2}$

Explanation for the correct answer:

Simplifying the given expression using trigonometric identities:

$$\begin{gathered} \sin \left[\left(\frac{\pi }{2}\right)-{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=\cos {\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\left[\because \sin \left(\frac{\pi }{2}\right)=\cos \right] \\ =\cos {\cos }^{-1}\left(\sqrt{1–\frac{3}{4}}\right)\left[\because {\sin }^{-1}x={\cos }^{-1}\left(\sqrt{1-x^2}\right)\right] \\ =\sqrt{1–\frac{3}{4}} \\ =\sqrt{\frac{4-3}{4}} \\ =\sqrt{\frac{1}{4}} \\ =\frac{1}{2} \end{gathered}$$

Thus, $\sin \left[\left(\frac{\pi }{2}\right)-{\sin }^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]=$$\frac{1}{2}$

Therefore, option (C) is the correct answer.