This answer supposes the OP actually wants to plot $\frac{1}{\sin(x)}$, not $\arcsin(x)$.
HINT: what is the value of the sine function on $\dots, -2\pi, -\pi, 0,\pi, 2\pi, \dots$ and on $\dots, -3\pi/2,-\pi/2,\pi/2,3\pi/2,\dots$? What's its behavior in between these points (positive/negative, increasing/decreasing)?
This is a craftman's hint to the question, which exploits the fact that we know exactly how $\sin(x)$ behaves.
If we are more clever, we can also exploit another fact of the sine function, the fact that it is periodic. You know that $\sin(x+2\pi)=\sin(x)$ for all $x\in \mathbb{R}$. This allows you to plot the sine function just on $[0,2\pi)$ and then "copy it" appropiately to get the graph on all $\mathbb{R}$.
We can exploit this in this case too, since $\frac{1}{\sin(x+2\pi)}=\frac{1}{\sin(x)}$ whenever $\sin(x)$ doesn't vanish. This means that, to plot $\frac{1}{\sin(x)}$, you might just as well plot it on the points of $[0,2\pi]$ where it is defined, and then copy the graph appropiately. This might make it easier, and puts on paper what you surely observed the moment you looked at the graph, that is, it is the same on $[-2\pi,0]$ and on $[0,2\pi]$.
There is another approach which is more mechanical and uses calculus:
HINT: Find out where the function is defined. At the points where it is not defined, find out the lateral limits. Now what calculus tool lets you find out if a function is increasing/decreasing? Compute it, and you will also find the local maxima/minima.