Math help : how do I solve out sin (u+v) + sin(u-v)?

My question pretty much says it all. I’m stranded on this very problem. The answer is supposed to be 2sinu x cosv and I just don’t understand how we get the last answer.
Simplify : Sin ( u + v) + Sin (u – v)

Sin ( u + v ) = sin u * cos v + cos u * sin v
Sin ( u – v ) = sin u * cos v – cos u * sin v
– sin u * cos v + cos u * sin v + sin u * cos v – cos u * sin v
– 2sin u * cos2v * sin2v
this is as far as I can go, I cannont figure out how the cos v comes in the answer. Please help, I’ve got a lot of these types of exercises awaiting me ahead.