Answer
$A$
Work Step by Step
The cosecant function is undefined when $x=..., -\pi, 0, \pi, 2\pi, ...$.
Thus, its graph has the vertical asymptotes $x=..., -\pi, 0, \pi, 2\pi, ...$.
RECALL:
The graph of the function $y=\csc{(x-d)}$ involves a $|d|$phase (horizontal) shift of the parent function $y=\csc{x}$.
The shift is to the right when $d\gt0$ and to the left when $d\lt0$.
The given function has $d=-\frac{\pi}{2}$ so it involves a phase shift of $\frac{\pi}{2}$ to the left.
Thus, its vertical asymptotes are $x=..., -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, ...$
The only possible graph among the choices are the ones in Options A and C.
Note that when $x=0$ , the value of $y=\csc{(x+\frac{\pi}{2})}$ is $1$.
Only the graph in Option $A$ satisfies this.
Thus, the graph of $y=\csc{(x+\frac{\pi}{2})}$ must be the one in Option $A$.
