I'm not entirely sure what you're looking for but when I worked this out I got:
cos(u)cos(v)sin(v) + sin(u)sin(v)^2 + sin(u)cos(v)^2 - sin(v)cos(u)cos(v) = sin(u)(cos(v)^2 + sin(v)^2) = sin(u)
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.