I think that the tidiest way to do this is to use the addition formulas

Use those formulas to get

(from (2)),

(from (1)),

and therefore

(subtracting (5) from (4)).

Also,

(from (3)).

Finally, substitute the result from (7) into (6) to get your formula.

The interesting thing here is that although the formula is completely symmetrical in , and , it is necessary to break the symmetry in order to get an efficient proof (in this case, by treating the term differently from its "partners" and ).