For all real numbers x , cos(2x)

Answer 1

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

It's a double angle formula

cos(2(45) = cos90 = 0 = cos^2(45) - sin^2(45) = (sqr2/2)^2 - (sqr2/2)^2 = 2/4 - 2/4 = 0

cos(2(30)) = cos60 = 1/.2 = cos^2(30) - sin^2(30) = (sqr3/2)^2 - (1/2)^2 = 3/4 -1/4 = 1/2

it works for any real number x

another double angle formula is

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

or

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

use

a trig identity is cos^2(x) + sin^2(x) = 1 or sin^2(x) = 1-cos^2(x) to derive them

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