Is there a way to solve for $x$ in $\cos(ax)/\cos(bx)=c$?

Question

Is there a way to solve for x in $\dfrac{\cos(ax)}{\cos(bx)} = c$?

This is similar to the question on $\dfrac{\cos^{-1}(ax)}{\cos^{-1}(bx)} = c$.

Answer 1

The answer is similar to the other one.

If $a/b = m/n$ is rational then the equation can be written as $\cos (m\theta) = c \cos (n \theta)$ where $ax = m \theta$. For $c = \pm 1$, $\theta$ will be an easily described angle, and if $|c| \neq 1$, then $\cos(\theta)$ is an algebraic function of $c$ since the equation is equivalent to $P_m(\cos \theta) = P_n( \cos \theta)$ where the $P$ are Chebyshev polynomials. If $c$ is an algebraic number then so is $\cos(\theta)$.

For $a/b$ irrational and $c \neq \pm 1$ there is no algebraic solution.