Here's the most important tip, for all such problems: WORK ON BOTH SIDES!!!!!!!!!!!

Sure, you are usually required to work on just one side, but working on both sides is like putting one finger on the "start", and one finger on the "end" of a maze, and finding a way to get your fingers to touch. Once you've found that path in the maze, you can retrace it from start to finish. The same goes with trig identities.

So I would multiply both sides by tan/tan, using tan = sin/cos where necessary. This gives...

cos/(1 + sin) = (1 - sin)/cos

Cross multiplying gives the usual identity.

So can you find a way to work this on one side only?

The easiest way that I can think of would be to multiply the left by tan/tan, then use conjugates, then multiply by cot/cot (all on the left).