Question:
Which of the following is the principal value branch of $\operatorname{cosec}^{-1}$ ?
(a) $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
(b) $[0, \pi]-\left\{\frac{\pi}{2}\right\}$
(c) $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
(d) $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$
Solution:
The principal value branch of $\operatorname{cosec}^{-1} x$ is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]-\{0\}$.
Hence, the correct answer is option (d).
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