Which points in the $x$-$y$ plane satisfy $\sin y=\sin x$?

Firstly, we know that the equation is trivially true when $x=y$, and so we can rule out graphs (b) and (d) as the line $y=x$ is missing.

Secondly, we know $y = \pi - x$ is a solution to the equation, and this line does not appear in (d), which means that by elimination, (c) is the answer.

A fuller answer might run as follows: we know that for all values of $x$, $$\sin(x)=\sin(\pi-x)$$ and $$\sin(x)=\sin(x + 2n\pi),$$ where $n$ is an integer.

Thus if $\sin(y) = \sin(x)$ then $$y = x + 2n\pi,$$ OR $$y = \pi - x + 2n\pi = -x +(2n-1)\pi.$$

So the solution is given by the lines $$\cdots, y=x-2\pi, y = x, y = x+2\pi, y = x + 4\pi, \cdots,$$ together with the lines $$\cdots, y = -x-\pi, y=-x+\pi, y=-x+3\pi, \cdots.$$

These are the lines shown in (c).