Existence of a solution for non-linear system of equations

Question

Given the two non-linear equations $$2\sin(x)-2\sin(a-x)+\sin(a)=0,\\ 2\cos(x)-2\cos(a-x)+\cos(a)-1=0, $$ where $0<a<2\pi$ and $0<x<a/2$. Is there a way to show that for arbitrary $a$, $x$ always exits such that both equations hold (I don't need an explicit expression for $x$, just its existence)?

Answer 1

The two equations can be expanded as $$-2\sin a\cos x+2(1+\cos a)\sin x+\sin a=0,\\ 2(1-\cos a)\cos x+2\sin a\sin x+\cos a-1=0$$

or

$$-2\cos x+2\frac{1+\cos a}{\sin a}\sin x+1=0,\\ -2\cos x-2\frac{\sin a}{\cos a-1}\sin x+1=0.$$

The rest is yours.