how to write cos 2x in terms of cos x?

We'll write cos 2x as the cosine of the sum of 2 like
angles:

cos(x+x) = cos x*cos x - sin x*sin
x

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

We'll write sin x in terms of cos x, applyingthe
fundamental formula of trigonometry:

(sin x)^2 + (cosx)^2 =
1

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

We'll substitute (2) in
(1):

cos(x+x) = (cos x)^2 - [1 - (cos
x)^2]]

We'll remove the
brackets:

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

We'll combine like
terms:

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

So,the expression of cos 2x, written in terms of cos x,
is:

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