For the principal value branch, the value of $\sin\left(\frac{\pi}{2}-\sin^{-1}(-\frac{\sqrt{3}}{2})\right)$ is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → $\frac{1}{2}$
Given expression: $\sin\left(\frac{\pi}{2}-\sin^{-1}(-\frac{\sqrt{3}}{2})\right)$
Use identity: $\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$
$= \cos\left(\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right)$
Let $\theta = \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)$
Then, $\cos(\theta) = \sqrt{1 - \sin^2(\theta)} = \sqrt{1 - \left(-\frac{\sqrt{3}}{2}\right)^2} = \sqrt{1 - \frac{3}{4}} = \sqrt{\frac{1}{4}} = \frac{1}{2}$