The smallest positive period of the function f(θ) = si$n^2$θ is:

sin(θ + 180°) = −sinθ, and squaring erases the minus sign, so $f$(θ + 180°) = $f$(θ). Equivalently si$n^2$θ = (1 − cos2θ)/2, and cos2θ repeats when 2θ grows by 360°, i.e. when θ grows by 180°. Nothing smaller works: $f$(0°) = 0 while $f$(90°) = 1, which rules out 90°. Quoting the sine's own period 360° ignores the squaring.