I have this equation where $f,g,h$ are real parameters and $0<x,y<2\pi$.
$$ f \,(\cos x)+ g\, (\cos y)= h$$
Using the identities $ \mid\cos x \mid<1 $ and $ \mid\cos y \mid<1 $, are we able to find a constraint over the parameters $f,g,h$ only? I mean can we obtain an inequality condition that contains only $f,g,h$?
We are looking for a sentence equivalent to
$$\exists x \exists y \ \ s.t. \ \ f * \cos x+ g * \cos y= h \tag{1}$$
as we don't know the signs of $f, g, h$, the extreme points of the LHS are to be taken among
$$A:=f+g, \ \ B:=f-g, \ \ C:=-f+g, \ \ D:=-f-g $$
$$\text{and} \ \ M:=max(A,B,C,D)\tag{2}$$
An inequality between $f,g,h$ equivalent to (1) is :
$$|h| \le M \ \ \iff \ \ -M \le h \le M \tag{3}$$
Remark: in fact $-M=min(A,B,C,D)$.
Proof of (3):
$$|f * \cos x+ g * \cos y\| \le |f| * |\cos x|+ |g|* |\cos y | \le |f|+ |g| = max(A,B,C,D)\tag{4}$$
Remark: one could say a little more about the existence of a solution in $x,y$, given $f,g,h$ when these parameters fulfill (3) using the continuity of the expression present in the LHS of (1) and Intermediate Value Theorem, but this is not asked...