How to solve equation involving Chebyshev polynomial ratio?: $\frac{\cos\big(\frac{\cos^{-1}(ya)}{2N}\big)}{\cos\big(\frac{\cos^{-1}(y)}{2N}\big)}=x$

Question

I have the following equation: involving the ratio of two Chebyshev polynomials:

$$\frac{\cos\left(\frac{\cos^{-1}(ya)}{2N}\right)}{\cos\left(\frac{\cos^{-1}(y)}{2N}\right)}=x$$

(for some reason, I am unable to type equations here even though I did exactly as the tutorials told me to)

where $a$ is an constant

I am unable to get the equation in terms of $x$. I couldn't find any identity involving the ratio of two chebyshev polynomials and couldn't think of any useful approaches.

Is there any way to get the equation in terms of $x$?