Which of the following is the principal value branch

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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