Prove the identity: cos (2x) = cos^2(x) - sin^2(x)?

Answer 1

cos(2x) = cos(x+x)

by the formula for cosine of sum of two angles, which you probably have already learned,

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

also, because cos^2(x) + sin^2(x) = 1 by the pythagorean identity, cos(2x) = 1-sin^2(x) - sin^2(x) = 1-2sin^2(x)
by similar reasonining, cos(2x) = 2cos^2(x) - 1

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