cos(u-v)sinv + sin(u-v)cosv
cosu*cosv + sinu*sinv + sinu*sinv^2 + sinu*cosv^2 - cosu*sinv*cosv
sin(any variable)^2 + cos(any variable)^2 = 1
This is the most important trig identity you'll ever learn. I'd prove this for you, but its 3 AM for me so here: Proofs of trigonometric identities - Wikipedia, the free encyclopedia.