Expert: Steve Holleran - 5/8/2007

Question
hi ,can you help me show that cos3x = 4cos^3x - 3cosx

thanks.

Answer
Hi Jay,

Okay, to do this, we'll use the addition formula for
cos(A+B), the identity cos 2x = 2 cos^2 x - 1, the identity sin 2x + 2 sin x cos x, and the pythagorean identity sin^2 x = 1 - cos^2 x :

cos(3x) = cos(2x + x)

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

       = 2 cos^3 x - cos x - 2 sin^2 x * cos x

       = 2 cos^3 x - cos x - 2(1-cos^2 x) * cos x

       = 2 cos^3 x - cos x - 2(cos x - cos^3 x)

       = 2 cos^3 x - cos x - 2 cos x + 2 cos^3 x

       = 4 cos^3 x - 3 cos x.

Hope this helps you out.

Steve Holleran