cos^2(x) = (1 + cos(2x)) / 2
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To prove this identity, we can start with the double-angle identity:
cos(2x) = cos^2(x) – sin^2(x)
We can rearrange this equation to solve for sin^2(x):
sin^2(x) = cos^2(x) – cos(2x)
Next, we can substitute this expression for sin^2(x) into the right-hand side of the original equation:
(1 + cos(2x)) / 2
= (1/2) + (cos(2x) / 2)
= (1/2) + (cos^2(x) – sin^2(x)) / 2 (using the above double-angle identity)
= (1/2) + (cos^2(x) – (cos^2(x) – cos(2x))) / 2 (using the derived expression for sin^2(x))
= (1/2) + (cos(2x) / 2)
= cos^2(x)
Thus, we have proven that cos^2(x) = (1 + cos(2x)) / 2.
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