The value of \(\sin^{-1}\left(-\displaystyle\frac{1}{2}\right) + \cot^{-1}(-1)\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{7\pi}{12}\)
The principal value branch of \(\sin^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\). For a negative number the angle is negative:
\( \displaystyle \sin^{-1}\left(-\frac{1}{2}\right) = -\frac{\pi}{6} \)
The branch of \(\cot^{-1}\) is \((0, \pi)\). For a negative number the angle is in the second quadrant:
\( \displaystyle \cot^{-1}(-1) = \pi - \frac{\pi}{4} = \frac{3\pi}{4} \)
Adding:
\( \displaystyle -\frac{\pi}{6} + \frac{3\pi}{4} = \frac{-2\pi + 9\pi}{12} \)
\( \displaystyle = \frac{7\pi}{12} \)
Why the other options are incorrect
Option (1) \(\frac{11\pi}{12}\) — Incorrect. This is \(\frac{\pi}{6} + \frac{3\pi}{4}\): it takes \(\sin^{-1}\left(-\frac{1}{2}\right)\) as \(+\frac{\pi}{6}\), but \(\sin\frac{\pi}{6} = +\frac{1}{2}\).
Option (3) \(-\frac{5\pi}{12}\) — Incorrect. This is \(-\frac{\pi}{6} - \frac{\pi}{4}\): it takes \(\cot^{-1}(-1)\) as \(-\frac{\pi}{4}\), but negative angles are outside \((0, \pi)\).
Option (4) \(\frac{5\pi}{12}\) — Incorrect. This is \(\frac{\pi}{6} + \frac{\pi}{4}\): both minus signs are ignored.