You could use the formula sin – sin = 2cos(half the sum)*sin(half the difference), but I don't know that that looks any better:
Thanks, Opalg. Turns out it didn't do anything special. A former student of mine came to me (she thinks I'm a genius..poor girl) to help her and that was the best I could come up with.
She came back later and told me her instructor said that all she had to do was write it without the double and triple angles. Duh! It doesn't simplify down to anything remotely simple or exotic.
Oh well. Thanks for taking the time to look at it.
~Dale