Can we obtain a sufficient constraint over the parameters $A$, $B$, and $C$ in the given equation? I have this equation
$$ A\;\big(\cos (x-y)+\cos x+\cos y\big)+B\;\big(\sin (x-y)-\sin x+\sin y\big)+C=0,  $$
where $-\pi\leq x,y \leq \pi$, and $A$, $B$, and $C$ are some real constants.
Then, according to the ranges of the trigonometric functions, is there a sufficient constraint over the parameters $A$, $B$, and $C$ to have the real solutions? If so, how can we obtain that condition?

Using Cauchy-Schwarz inequality repeatedly: $C^2 = (A\cos(x-y)+B\sin(x-y) +A\cos x-B\sin x + A\cos y + B\sin y)^2 \le 3(m^2+p^2+q^2), m^2 \le A^2+B^2, n^2 \le A^2+B^2, p^2 \le A^2+B^2\implies C^2 \le 3(A^2+B^2+A^2+B^2+A^2+B^2)= 9(A^2+B^2). $, Thus the necessary condition is: $C^2\le 9A^2+9B^2$.
