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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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