First thing I would do is take the sine of both sides, and use the addition of angles formula on the RHS.
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!