sin^2x (Power to Double Angle)
1/2(1-cos2x)
To find the double angle formula for sin^2x, we make use of the identity:
sin(2x) = 2sin(x)cos(x)
We can then rearrange the terms as follows:
sin^2(x) = (1-cos^2(x)) (using the identity sin^2(x) + cos^2(x) = 1)
sin^2(x) = 1 – cos^2(x)
Now, we replace x with 2x in the above expression and simplify:
sin^2(2x) = 1 – cos^2(2x)
sin^2(2x) = 1 – (cos^2(x) – sin^2(x))^2 (using the identity for cos^2(2x))
sin^2(2x) = 1 – cos^4(x) + 2sin^2(x)cos^2(x)
sin^2(2x) = 2sin^2(x) – (2sin^2(x)cos^2(x))
sin^2(2x) = 2sin^2(x)(1 – cos^2(x))
sin^2(2x) = 2sin^2(x)sin^2(x) (using the identity 1 – cos^2(x) = sin^2(x))
sin^2(2x) = (sin^2(x))^2
Therefore, the double angle formula for sin^2(x) is:
sin^2(2x) = (sin^2(x))^2
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