Motion of a particle in x - y plane is described by a set of...

$x = 4\sin \left(\dfrac{\pi}{2} - \omegat\right) = 4\cos \left(\omegat\right)$,

$y = 4\sin \left(\omegat\right)$

Clearly, ${x}^{2} + {y}^{2} = {\left(4\cos \left(\omegat\right)\right)}^{2} + {\left(4\sin \left(\omegat\right)\right)}^{2} ⇒ {x}^{2} + {y}^{2} = 16$

Hence, the path covered by particle is a circle.