Chapter 4 - Section 4.6 - Graphs of Other Trigonometric Functions - Exercise Set - Page 610: 38

Please see below.

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 $y=-\frac{1}{2}\csc \pi x$ is $\frac{2\pi }{\pi }=2$.