Finding parametrized solutions to a trigonometric equation I want to find what pairs $(x,y)$ satisfy $\cos(x-y) = \cos(x) - \cos(y)$. To start, I defined a function $f(x,y) = \cos(x-y) - \cos(x) + \cos(y)$. I want to find functions $x(t), y(t)$ s.t. $f(x(t), y(t)) = 0$ I didn't really know where to go, so I tried putting the cosines in exponential form:
$$
\begin{align}
0 &= \cos(x-y) - \cos(x) - \cos(y)\\
0 &= \frac{1}{2}\big[e^{i(x-y)}+e^{i(y-x)}-e^{ix}-e^{-ix}+e^{iy}+e^{-iy}\big]\\
\end{align}
$$
From here I wasn't sure what to do. I tried multiplying through to get rid of negative exponents, rearranging into a neater format, factoring, etc. to no avail. I'm not even sure that's a useful first step, it just was the only one I could think of.
Any tips or ideas?
Also feel free to add or edit tags, wasn't quite sure what to put this under.

I'm not sure parametrizing the solution is a good idea, if you take a look at wolframalphas sketch of all solutions you'll see that x(t) and y(t) wouldn't even be continous. Maybe proving that all pairs of (x,y) is on a certain form would be a simpler solution.
