# How to solve this equation by Solve?

I have an equation to be solved. But Mathematica does not work for it. I hope the solution x can be expressed as a function of a and b

Solve[(x/Sin[a])^2 == ((1 - x)/Sin[b])^2 + 1 + (x/Tan[a] - (1 - x)/Sin[b] Cos[b])^2 - 2 ((1 - x)/Sin[b]) Sqrt[1 + (x/Tan[a] - (1 - x)/Sin[b] Cos[b])^2] Sin[b - ArcTan[x/Tan[a] - (1 - x)/Sin[b] Cos[b]]], x]

By using $$x = \frac{y}{l}$$, the above equation is reduced from the following one:

Solve[(y/Sin[a])^2 == ((l - y)/Sin[b])^2 + l^2 + (y/Tan[a] - (l - y)/Sin[b] Cos[b])^2 - 2 ((l - y)/Sin[b]) Sqrt[l^2 + (y/Tan[a] - (l - y)/Sin[b] Cos[b])^2] Sin[b - ArcTan[(y/Tan[a] - (l - y)/Sin[b] Cos[b])/l]], y]

If the reduced equation can be simplified to zero, how about this original one? I cannot simplify the original one to zero. From this original equation, can we get y as a function of l, a and b?

• Mathematica solves your equation and gives {}: no solution! – Ulrich Neumann Jul 1 '19 at 20:47
• Yes, why is there no solution? In principle, there should be a solution x as a function of a and b. I do not know why Mathematica gives { }. – Hao Wu Jul 1 '19 at 20:50
• Solve probably fails to get a "proper" simplification. It is actually tautologically true (that is, for all values of x). – Daniel Lichtblau Jul 1 '19 at 20:56

(x/Sin[a])^2 == ((1 - x)/Sin[b])^2 + 1 + (x/Tan[a] - (1 - x)/Sin[b] Cos[b])^2 -