Solving a simple trigonometric equation $\sin x = -\sin y$

Question

What is the solution set of the following trigonometric equation?

$$\sin x = -\sin y$$

Answer 1

Hint: $${-\sin(x)=\sin(-x).}$$

Answer 2

$\sin(\pi+x)=-\sin(x)$ Also $\sin(-x)=-\sin(x)$

Answer 3

The problem is equivalent to $$\sin(x)+\sin(y)=0$$ which is $$\frac{1}{2}\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)=0.$$ So, the general solution is $$x+y=2n\pi,\ \text{or}\ x-y=(2m+1)\pi$$ where $m$ and $n$ are integers.