Let's focus on simpler curves first.
(cos t, sin t): unit circle
(cos t, -sin t): y coordinate reflected => unit circle
(cos t, 1-sin t): y coordinate shifted up by 1 => unit circle shifted up
((4/5)cos t, 1-sin t): x coordinate multiplied by 4/5 => ellipse
The last coordinate (-3/5)cos t is just a multiple of the first coordinate => ellipse in space in the plane given by (4/5)x = (-3/5)z.
So finding the trace of the curve translates into describing what shape it represents? Do I have to prove it's an ellipse somehow?
Not sure what it means.
If you want to find an equation observe x2 + y2 = cos(t)2 + sin(t)2 = 1 is the equation for a circle. So the equation in the xy coordinates is (5/4 x)2 + (1 - y)2 = 1, and the second equation is (4/5)x = (-3/5)z. Visually, it's the intersection of a compressed cylinder in the z direction with that plane.
Thank you very much, only a few hours left to turn it in and you are a savior.