The range of the principal value branch of \(\mathrm{cosec}^{-1} x\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\left[-\displaystyle\frac{\pi}{2}, \displaystyle\frac{\pi}{2}\right] - \{0\}\)
\(\mathrm{cosec}^{-1} x = \theta\) means \(\mathrm{cosec}\,\theta = x\). Since \(\mathrm{cosec}\,\theta = \frac{1}{\sin\theta}\), the angle must have \(\sin\theta \ne 0\).
The principal value branch uses the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), on which sine takes each value once. Only \(\theta = 0\) has to be removed:
\( \displaystyle \sin 0 = 0, \quad \text{so } \mathrm{cosec}\, 0 \text{ is not defined} \)
The two ends are taken:
\( \displaystyle \mathrm{cosec}^{-1}(1) = \frac{\pi}{2}, \quad \mathrm{cosec}^{-1}(-1) = -\frac{\pi}{2} \)
Hence the range is \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right] - \{0\}\).
Why the other options are incorrect
Option (1) \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\) — Incorrect. It contains 0, but \(\mathrm{cosec}\,0\) is not defined, so no x has \(\mathrm{cosec}^{-1} x = 0\).
Option (3) \([0, \pi] - \left\{\frac{\pi}{2}\right\}\) — Incorrect. This is the range of the principal value branch of \(\sec^{-1} x\).
Option (4) \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\) — Incorrect. This is the range of \(\tan^{-1} x\); it leaves out \(\pm\frac{\pi}{2}\), which are taken at \(x = \pm 1\), and it contains 0.