Having these 4 equations, which are having 4 unknown variables ( x, y, α, and β ) as follows:
$$ \left\{ \matrix{ x^{\,2} - y^{\,2}= A^{\,2} + B ^{\,2} - 2AB \cos \left( {\alpha - Δ } \right) \hfill \cr A\cos \left( {\alpha - \beta } \right) = B \cos \left( {\beta - Δ } \right) \hfill \cr y\sin \beta +x = C - A \hfill \cr y\cos \beta = - D \hfill \cr } \right. $$
What would be the value of x in terms of "known" values A,B,C,D, and Δ ?
This set of equations is the result of this solution
The trigonometric functions make this a bit of a pain. So I would recommend getting rid of them.
First get rid of the angle differences in the arguments. https://en.wikipedia.org/wiki/List_of_trigonometric_identities#Angle_sum_and_difference_identities has formulas for turning each (trigonometric function of (sum or difference)) into (product of (trigonometric functions of the individual variables)). So that way you only have trigonometric functions of single variables.
Secondly, turn trigonometric functions of unknown variables into rational functions using https://en.wikipedia.org/wiki/Tangent_half-angle_formula. This maps angles to real values, with one exception namely the angle $\pm\pi$ would map to $\infty$. You might want to cater for that case separately. Given that you have multiple variables as arguments of trigonometric functions, that would be an annoying number of special cases, though. Don't do anything to the trigonometric functions of known quantities; these are just numbers after all.
Then multiply each equation by the common denominator of all the rational functions contained in them. This will turn the whole equation from a rational equation to a polynomial equation. So at this stage you have a system of four polynomial equations.
https://en.wikipedia.org/wiki/Elimination_theory has some pointers on how to solve such a system. Gröbner bases are one common approach, resultants another. Given the likely degrees of your equations, I'd recommend getting a computer algebra system to do the heavy lifting for you. Essentially all of these will combine equations in ways that eliminate some of the unknowns, so that you end up with one equation that only has a single unknown. You solve for that, then trace back to derive the others if you need them.
Perhaps factor equations during step 4, so that instead of a big equation of high degree you can deal with multiple much smaller equations of lower degree as separate cases.
If you do use a computer algebra system eventually, you can give it a try and see what it does for the initial set of equations. Perhaps hunting that all the variables will be real numbers. It will depend a lot on the quality of the computer algebra system on whether it will be able to find solutions, and whether you can trust it to have found all of them. Which is the reason why I'd still do it the above manual way: the theory of solving polynomial equations is so much better understood than that for general equations.
In your specific case, the trigonometric functions come from the fact that that one answer used polar coordinates to describe certain vectors. My answer to the same question used Cartesian coordinates, which avoided introducing all those trigonometric functions in the first place. You still end up with a system of four polynomial equations, and you still need elimination theory to solve that, but the expressions should be no worse, and most likely simpler, than the ones you get from a prior detour via the polar representation. I have updated my answer to link to the resulting factor equations, so you get an idea of what those look like.
