SOLUTION: Verify the following identity by using an angle sum identity: cos (2x) = 1 � 2(sin2 x). Hint (2x = x + x)

Question 890642: Verify the following identity by using an angle sum identity: cos (2x) = 1 � 2(sin2 x). Hint (2x = x + x)
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you want to prove the identity:

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

your basic identity for cos(x + y) is:

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

let y = x, and this basic identity becomes:

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

simplify to get:

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

another basic identity is sin^2(x) + cos^2(x) = 1

from this basic identity, we can solve for cos^2(x) to get:

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

replace cos^2(x) with 1 - sin^2(x) in your equation of:

cos(2x) = cos^2(x) - sin^2(x) to get:

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

simplify this to get:

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

since that is equal to your original equation, you are done.

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