how do your right a double argument property of cos (2x)?

Question

do you just sub it into the formula cos (a+b)= cos A cos B - sin A sin B

and then how would you express it into terms of cos x?

Answer 1

you're on the right track

cos(2x) = cosx cosx - sinx sinx
= cos^2(x) -sin^2(x)

more commonly it's written in one ofthe following two forms by using the expression:
sin^2(x) + cos^2(x) = 1

1. get rid of the cos^2(x) by 1 - sin^2(x) to give
cos(2x) = 1 - sin^2(x) - sin^2(x) = 1 -2sin^2(x)

2. similarly get rid of the sin^2(x) to get
cos^2(x) - sin^2(x) = 2cos^2(x) -1


An interesting exercise for you to do is to figure out how much cos(x/2) and sin(x/2) are in terms of sin(x) and cos(x)

Answer 2

cos 2x
= cos² x - sin²x
= 1 - 2sin²x

= 1 - 2.(1 - cos²x)
= 2cos²x - 1