1. Inverse Trigonometric Functions

Here I have a trigonometric equation, where a, A, b, and B are constants. This equation needs to be solved for x.

0 = a*cos(Ax)+b*cos(Bx)

If the second equation below could be solved for x, that would also solve the problem. Solving one will lead to a solution for the other.
I will use different constants than above to avoid confusion.

G*x = arccos(H*cos(x))

2. I'm assuming we're solving for coefficients $a$ and $b$.

We know $arccos(cos(x)) = x$.

Solve the second equation and plug the answer into the first equation.

After you simplify shift one of the cos terms to the other side and determine what the relationship is.

3. I'm sorry, I suppose that when I tried to make a quick question I just made it sloppy and unspecific. I'll try to clean my question up a bit.