What would be the correct approach solve this complicated equation in Mathematica?

Questions
1

I am just getting started with Mathematica and I have become familiar with quite a few basic operations. I wanted some advice on solving a complicated equation which I have encountered.

This is the equation I am trying to solve for \alpha1^{\circ}:

\text{c1} \left(\frac{1}{\cos (\text{$\alpha $1} {}^{\circ}) \sqrt{1-\text{M1}^2 \sin ^2(\text{$\alpha $1} {}^{\circ})}-\text{M1} \sin ^2(\text{$\alpha $1} {}^{\circ})}+\text{M1}\right) = \text{$\alpha $2} (\text{M2}+1)

In this equation, M1, M2, c1 and c2 are all non-zero and non-negative constants.
I have worked out by hand the solution of this equation but I want to understand how I can do this in Mathematica. This is the approach which I have used:

  1. In the complicated equation let
Y = \cos (\text{$\alpha $1} {}^{\circ}) \sqrt{1-\text{M1}^2 \sin ^2(\text{$\alpha $1} {}^{\circ})} - \text{M1} \sin ^2(\text{$\alpha $1} {}^{\circ})

This then simplifies the equation of interest to

\text{c1} \left(\text{M1}+\frac{1}{\text{Y}}\right) = \text{$\alpha $2} (\text{M2}+1)
  1. The last simplified equation can be solved for Y which yields
Y = \frac{\text{c1}}{-\text{c1} \text{M1}+\text{c2} \text{M2}+\text{c2}}

Since the solution Y is composed entirely of constants, the final solution will also be a constant. Thus, let

K = \frac{\text{c1}}{-\text{c1} \text{M1}+\text{c2} \text{M2}+\text{c2}}

With this simplification the equation of interest becomes Y = K Substituting the simplified form of Y from Step 1) leads to

\cos (\text{$\alpha $1} {}^{\circ}) \sqrt{1-\text{M1}^2 \sin ^2(\text{$\alpha $1} {}^{\circ})} - \text{M1} \sin ^2(\text{$\alpha $1} {}^{\circ}) = K
  1. In Y, let x = \sin^{2}(\alpha1) then Y in terms of x becomes
Y = \sqrt{1-x} \ \ \sqrt{1-\text{M1}^2 x}-\text{M1} x
  1. The last equation can be solved for x and after substituting the relevant expressions the final answer will be given in terms of \sin^{-1} and there will be four solutions of \alpha1.

These are the four major steps I have used to manually solve the equation. Can someone please advise on what would be right approach to solve this in Mathematica ? What would be the right approach? I have attached here the notebook with the LHS and RHS parts of the equation typed in.

Open in Wolfram Cloud ↗
2

I think (1-x^2) should actually be sqrt(1-x) near the end.

3

Yes. I corrected it.

4

I get four solutions with Solve this way:

sols = Solve[LHS == RHS, \[Alpha]1]

The solutions are to be verified. Here is how they behave with a choice of the parameters:

paramValues = {c1 -> 1, c2 -> 1, M1 -> 2, M2 -> 3};
sols /. paramValues
% // N
LHS == RHS /. sols /. paramValues