# Thread: What is the process to prove these Identities

1. ## What is the process to prove these Identities

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)

2. Originally Posted by yeunju
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)
This looks like a half angle fromula with 2x instead of x, so try a half angle identity.

#2
sin(x + pi/2)= cos x
Since you have the sign of a sum try the sum identity for sins

#3
tan^2(x) = sec^2(x)= sin^2(x) - cos^2(x)
replace tan and sec with their definitions in terms of sins and cosines and simplify

#4

cot(x)/(1 + csc(x)) = (csc(x) - 1)/ cot(x)
This is equivalent to:

cot^2(x)=(csc(x)+1)(csc(x)-1)=csc^2(x)-1

Now replace everything with its definition in terms of sins and cosins and simplify both sides.

CB