Answer
$x=2 \cos t ; y =2 \sin t$ and $z=2 \sin (2t)$; $0 \leq t \leq 2 \pi$
or,
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t$; $0 \leq t \leq 2 \pi$
Work Step by Step
The parametric equations of a circle having radius $r$ are; $x=r \cos t ; y =r \sin t$
Here, we have $x^2+y^2=4$ and radius is $r=\sqrt 4=2$
Thus, the parametric equations of a circle having radius $2$ are:
$x=2 \cos t ; y =2 \sin t$
and $z=xy=(2 \cos t) (2 \sin t)= 4 \sin t \cos t$
Hence, the parametric equation becomes:
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t= 2 [2\sin t \cos t]= 2 \sin (2t)$; $0 \leq t \leq 2 \pi$
or,
$x=2 \cos t ; y =2 \sin t$ and $z=4 \sin t \cos t$; $0 \leq t \leq 2 \pi$
