I'm all out of ideas here. Anyone?
I'd start with the Right-hand side. Multiply numerator and denominator by 1 + sin(x) and see what happens.
Like many such problems, it may not lead anywhere, but it's a place to start and something else may hit you.
For example, while I was typing this, it dawned on me that I don't happen to know any formulas for tan(2x) right off the top of my head. I might be tempted to change that to sines and cosines. Maybe not.
Rule #1 on Trig Identities -- You can't break it. Just try something.