Which of the following is the principal value branch of $\text{cosec}^{-1}x$?

Question

$\begin{array}{1 1}(A)\quad\bigg(\frac{-\pi}{2},\frac{\pi}{2}\bigg) & (B)\quad(0,\pi)-\left \{\frac{\pi}{2}\right \}\\(C)\quad\begin{bmatrix}\frac{-\pi}{2},\frac{\pi}{2}\end{bmatrix} & (D)\quad\begin{bmatrix}\frac{-\pi}{2},\frac{\pi}{2}\end{bmatrix}-\{0\}\end{array}$