Okay, second one:
sin^2(u)(cot^2(u)+1) -- distributive property
Then, change cot^2(u) to cos^2(u)/sin^2(u), use some algebra to un/re-distribute it, sin^2(u) should cancel on the first term (the fraction), and then use the identity cos^2(u)+sin^2(u) = 1
Third one, I guess you could use the sum/difference identities, but I don't know if that really counts as simplifying. (sin(u-v)=sin(u)cos(v)-cos(u)sin(v))
Fourth looks like a royal bastard and I don't see a way to do it easily off my head.
And fifth, it's been too long since I've done trig and it's too late for me to try too hard.