Prove that the cosine of double angle may be written in terms of sine of angle.

We'll write the cosine of double angle as cos 2x. We know
that the cosine of double angle can be written as 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 cos x in terms of sin x, applying the
fundamental formula of trigonometry:

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

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

We'll combine like
terms:

cos 2x = 1 - 2(sin
x)^2

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

cos 2x = 1 - 2(sin
x)^2