$\left(\because \cos 2 \theta=1-2 \sin ^{2} \theta\right) $

$= \frac{a}{2}-\frac{a \cos 2 \omega t}{2}$

$\therefore \quad$ Velocity, $v=\frac{d x}{d t}=\frac{2 \omega a \sin 2 \omega t}{2}=\omega a \sin 2 \omega t$

Acceleration, $a=\frac{d v}{d t}=2 \omega^{2} a \cos 2 \omega t$

For the given displacement $x=a \sin ^{2} \omega t,$

$a \propto-x$ is not satisfied.

Hence, the motion of the particle is non simple harmonic motion.