The RHS does not simplify as you have indicated. You have to use the addition of angles formula.
[EDIT] Also, don't forget the Commandments of Algebra.
Yeah, you skipped a few steps, but I would agree with your conclusion. You do need to be careful: in squaring the equation to end up with your quadratic, you may have introduced spurious solutions. So you should definitely double-check against your allowed x's. Good work!