Prove that (cosx-sinx)(cosx+sinx)=cos2x?

We'll manage the left side because we notice that the
special product from the left side returns the difference of two
squares:

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

But this difference of two squares represents the double
angle formula:

cos 2x = cos^2 x - sin^2
x

Another way to solve the problem is to remove the
brackets from the left side:

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

We'll
eliminate like terms:

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

We'll recognize the
identity:

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

Therefore, the given expression
(cosx-sinx)(cosx+sinx) = cos 2x represents an
identity.