A problem in Optics comes down to this equation:
Given that $A$, $B$, $C$ and $D$ are all real positive numbers, $-1 \lt x \lt 1$ and $-1 \lt \sqrt{C} x \lt 1$
$$\frac{Ax}{\sqrt{1 - x^2}} + \frac{Bx}{\sqrt{1 - Cx^2}} = D$$
How to solve this kind of equation?
On
Firstly I think that either the second denominator is $\sqrt{1-(Cx)^2}$ or the condition with $C$ is $-1 \leq \sqrt{C}x \leq 1$ with the additional $C \geq 0$.
A possible approach to the equation is the substitution $x = \cos{y}$ ot $x = \sin{y}$ (since $|x|<1$). With the first substitution, the equation becomes $ \frac{B\cos{y}}{\sqrt{1-C\sin^2{y}}} = D - A\cot{y}$. Now a necessary condition for the equation to have a solution is $(D-A\cot{y})(B\cos{y}) \geq 0$. For all such $y$ rise to the power of two and you will get rid of the square root. However, the left trigonometric equation could be difficult to solve explicitly for arbitrary parameters $A,B,C,D$. You can use universal substitution to make the equation polynomial. If you obtain some roots remember check if for them holds $(D-A\cot{y})(B\cos{y}) \geq 0$. It is possible that the problem must be solved numerically after the last step with the universal substitution.
You multiply by the product of the denominators and square the result. That will get you down to one square root, which you can isolate and square again. Unfortunately, that gives you an eighth degree equation unless there is cancellation. You can do numeric root finding on the original.