cos(2x) = 1 - 2(sin(x)^2), so sin(x)^2 = 1/2 - 1/2*cos(2x).
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No.
Do sin(x), square it, and then multiply it by two.
Yes. Except where sin x = 0, because then you would be dividing by zero so the quotient is undefined.
The deriviative of sin2 x + cos2 x is 2 cos x - 2 sin x