Solve $\cos(a)+\cos(b)=\cos(c)+\cos(d)$ for $a, b, c, d\in(0, \pi)$.

Question

I am trying to solve this equation in variables, $a, b, c, d$, i.e., to see what conditions they must satisfy. Say that $a, b, c, d$ are in $(0, \pi)$.

Answer 1

For any $a, b, c$ such that $|\cos(a)+\cos(b)-\cos(c)|\leq 1$, you can find the necessary $d$ by $$ d= \cos^{-1}\left(\cos(a)+\cos(b)-\cos(c)\right) $$ For any other $a, b, c$, there is no solution.