Short: Mathematica can't
However: Wolfram Alpha might
Alternatively the following gives some insight. When you have an expression that is just the sum of sines and cosines and powers, generally converting to exponentials and (re)combining them gives simpler expressions:
-2 Sin[\[Beta]/2 - \[Gamma]/2] Sin[\[Beta]/2 + \[Gamma]/2]
TrigToExp[%]
ExpToTrig[%]
