Trig simplification issues

I'v been sick in my college math class and have been a bit behind. I really could use some help and have been slamming my head against my text book trying to figure some of these simplifications out.(Headbang) One in particular that's been driving me nuts is this.

cosx/(((cosx-1)/(cosx))-1) Sorry for my overuse of parentheses here not sure how to make it look all professional.

another is

if Asin(squared)x+Bcos(squared)x =1 show that

sin(squared)x=(1-b)/a-b) and tan(squared)x=(b-1)/(1-a)

my last one is

suppose Asinx+cosx=1 and bsinx-cosx=1 show that AB=1

any help you can offer would be much appreciated as I've been working on these problems for hours and solved nothing.