The principal value of cosec^-1(-2) is

The correct answer is Option (2) → \(-\displaystyle\frac{\pi}{6}\)

The principal value branch of \(\mathrm{cosec}^{-1}\) is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\). We need the angle in this branch whose cosecant is \(-2\).

Since \(\sin\frac{\pi}{6} = \frac{1}{2}\),

    \( \displaystyle \mathrm{cosec}\,\frac{\pi}{6} = 2 \)

Cosecant is an odd function, so

    \( \displaystyle \mathrm{cosec}\left(-\frac{\pi}{6}\right) = -2 \)

and \(-\frac{\pi}{6}\) lies in the branch. Hence \(\mathrm{cosec}^{-1}(-2) = -\frac{\pi}{6}\).

Why the other options are incorrect

• Option (1) \(-\frac{\pi}{3}\) — Incorrect. \(\mathrm{cosec}\left(-\frac{\pi}{3}\right) = -\frac{2}{\sqrt{3}}\), not \(-2\).

• Option (3) \(\frac{7\pi}{6}\) — Incorrect. \(\mathrm{cosec}\,\frac{7\pi}{6} = -2\), but \(\frac{7\pi}{6}\) is outside the principal branch.

• Option (4) \(\frac{5\pi}{6}\) — Incorrect. \(\mathrm{cosec}\,\frac{5\pi}{6} = +2\), and \(\frac{5\pi}{6}\) is outside the principal branch.