Answer
Please see below.
Work Step by Step
Please note that since $\csc (Bx-C)=\frac{1}{\sin (Bx-C)}$, the period of the cosecant function is the same as the period of the sine function, that is, $\frac{2\pi }{B}$.
So the period of the function $y=\csc (x-\frac{ \pi }{2})$ is $\frac{2\pi }{1 }=2\pi$.