The value of \(\sin\left[\displaystyle\frac{\pi}{2}-\sin^{-1}\left(-\frac{\sqrt{3}}{2}\right)\right]\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \(\frac{1}{2}\)
The principal value of \(\sin^{-1}\) lies in \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), so
\( \displaystyle \sin^{-1}\left(-\frac{\sqrt{3}}{2}\right) = -\frac{\pi}{3} \)
Then
\( \displaystyle \sin\left[\frac{\pi}{2} - \left(-\frac{\pi}{3}\right)\right] = \sin\left(\frac{\pi}{2} + \frac{\pi}{3}\right) \)
\( \displaystyle = \sin\frac{5\pi}{6} = \sin\left(\pi - \frac{\pi}{6}\right) \)
\( \displaystyle = \sin\frac{\pi}{6} = \frac{1}{2} \)