How do you prove cos2x = 2cos²x - 1?

We have to prove cos 2x = 2cos2x -1

Proof

Let us start from LHS

We know that cos 2x can be expressed as

cos 2x = cos (x +x)

cos(A+B)=cos(A)⋅cos(B)−sin(A)⋅sin(B)

Using the above identity we will solve cos (x + x)

cos 2x = cos (x + x) = cos x . cosx – sin x . sin x

cos 2x = cos (x + x) = cos2x – sin2x

We know the identity sin² a + cos² a = 1

so sin² a = 1-cos² a

cos 2x = cos (x + x) = cos²x – 1- cos² x

cos 2x = cos (x + x) = 2cos²x – 1

= RHS

Hence Proved