To prove:



Work with left side:



Change to sines and cosines:



Invert and multiply







Use a 'product to sum' formula, which is:

         
         [That's easy to prove using the formulas for sin(A±B)]







Use the "odd-even" identity sin(-A) = -sin(A)









Edwin

I tried it using the identity that you chose, but it became
more and more complicated, so I tried the "product to sum"
formula above, and, as you see, it was fairly straight forward.  
I could also have used the "sum to product" formula working 
with the right side.  The formulas you need for such problems are:

Sum or difference to product:









Product to sum or difference






 
All those are easily proved using formulas for sin(A±B) and cos(A±B)

Edwin