What must I do to plot the graph of $\sin x=\sin y$ I must represent the domain of the function: $$z=\frac{x-y}{\sin x-\sin y}$$ 
Therefore, $\sin x\neq\sin y$. 
So I must plot $\sin x=\sin y$. How do I do this?

$\sin y = \sin x \iff y = n\pi +(-1)^nx,\text{ for some $n\in \mathbb{Z}$}$.
Putting $n=0$, we see that $y = x$ is on the graph.
Putting $n=1$, we see that $y = -x +\pi$ is also on the graph.
So for all even $n$, we have the set of straight lines:
$y = x + -4\pi,\,\quad y = x + -2\pi,\,\quad y = x,\,\quad y = x + 2\pi$ and so on so forth.
Likewise, for all odd $n$:
$y = -x + -3\pi,\,\quad y = -x + -\pi,\,\quad y = -x + \pi,\,\quad y = -x + 3\pi$ and so forth.
