Answer
$y=1 - \csc{(\frac{1}{2}x)}$
Work Step by Step
The graph looks like a reflection about the x-axis of the graph of the basic cosecant function so its tentative equation is $y=-\csc{bx}+d$
RECALL:
The period of $y=-\csc{(bx)}+d$ is $\frac{2\pi}{b}$.
The period of the given graph is $4\pi$.
Thus,
$4\pi=\frac{2\pi}{b}
\\4\pi(b) = 2\pi
\\\frac{4\pi(b)}{2\pi}= \frac{2\pi}{4\pi}
\\b=\frac{1}{2}$
This means that the tentative equation of the given graph is $y=-\csc{(\frac{x}{2})}+d$
Notice, that instead of having vertices whose y-coordinates are either $-1$ or $1$, the vertices have the y-coordinates $0$ and $2$.
This means that the given graph involves a 1-unit upward shift of the parent function $y=\csc{x}$.
Therefore, the equation of the function whose graph is given is $y=-\csc{(\frac{1}{2}x)} +1$ or $y=1 - \csc{(\frac{1}{2}x)}$.
