This is a very interesting problem. It has either too much symmetry, or not enough, depending on how you're looking at it.
One thing I would highly recommend: abbreviate! I'm going to abbreviate as follows:
If you square both equations and add together in the appropriate way, you can get it down to
Unfortunately, if you plug in a test case, you find out that the sin terms do not necessarily make up half of the sum. Here's one test case:
approximately. All of these are in radians. The sin terms add up to , and the cosine terms add up to
So there's a specific case, plus some general information about the problem. Nothing more is coming to mind at the moment. Your result should be -3/4. But now you have to prove it!