How would you determine $\sin(x) = -\cos(x)$ I can pretty much look at the equation and say the answer is $\frac{3\pi}{4} + 2\pi k$ and $\frac{7\pi}{4} + 2\pi k$ (where $k \in \mathbb N$) but how can I show work for this?

This may be overkill, but it could be useful in different situations, for instance something like $\sin x=-\cos3x$.
Recall that $-\cos x=\sin(x-\pi/2)$ so your equation becomes
$$
\sin x=\sin\left(x-\frac{\pi}{2}\right)
$$
This amounts to saying that either
$$
x-\frac{\pi}{2}=x+2k\pi
$$
(which has no solutions) or
$$
x-\frac{\pi}{2}=\pi-x+2k\pi
$$
that becomes
$$
2x=\pi+\frac{\pi}{2}+2k\pi
$$
and, finally,
$$
x=\frac{3}{4}\pi+k\pi
$$
