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

We know that, the principal value branch of ${{\mathop{\rm cosec}\nolimits} ^{ - 1}}x$ is $\left[ {\frac{{ - \pi }}{2},\frac{\pi }{2}} \right] - 0$.

$therefore,$ $y = {{\mathop{\rm cosec}\nolimits} ^{ - 1}}x$