# How do you prove #cos2A = 2cos^2 A - 1#?

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To prove that cos(2A) = 2cos^2(A) - 1, we can use the double angle identity for cosine:

cos(2A) = cos(A + A)

Using the cosine addition formula, cos(A + B) = cos(A)cos(B) - sin(A)sin(B), we have:

cos(2A) = cos(A)cos(A) - sin(A)sin(A)

Since cos(A)cos(A) is the same as cos^2(A) and sin(A)sin(A) is the same as sin^2(A), we can rewrite the equation as:

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

Using the Pythagorean identity sin^2(A) + cos^2(A) = 1, we can replace sin^2(A) with 1 - cos^2(A):

cos(2A) = cos^2(A) - (1 - cos^2(A))

Expanding the expression:

cos(2A) = cos^2(A) - 1 + cos^2(A)

Rearranging terms:

cos(2A) = 2cos^2(A) - 1

Therefore, cos(2A) = 2cos^2(A) - 1 is proven.

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