I would like some input in which identities to use for each problem. I'm never sure whether to use sum or differences, double angle, 1/2 angle, and etc..there are so many, so I guess and check, which isn't the smart way. I would like to know how to choose the identities.

#1

cos(4x) = 1 - 2sin^2(2x)

#2

sin(x + pi/2)= cos x

#3

tan^2(x) = sec^2(x)= sin^2(x) - cos^2(x)

#4

cot(x)/(1 + csc(x)) = (csc(x) - 1)/ cot(x)