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

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