2 sin^2 theta = 1/4
sin^2 theta = 1/8
sin theta = sqrt(1/8)
theta = arcsin(sqrt(1/8))
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If r-squared = theta then r = ±sqrt(theta)
cos2(theta) = 1 cos2(theta) + sin2(theta) = 1 so sin2(theta) = 0 cos(2*theta) = cos2(theta) - sin2(theta) = 1 - 0 = 1
Your question is insufficiently precise, but I'll try to answer anyway. "Sine squared theta" usually means "the value of the sine of theta, quantity squared". "Sine theta squared" usually means "the value of the sine of the quantity theta*theta". The two are not at all the same.
If sin (theta) is 3/5, then sin2 (theta) is (3/5)2, or 9/25.
It is 2*sin(theta)*sin(theta) because that is how multiplication is defined!