y = cos(2t), y = sin(2t);
t is greater than or equal to -pie and less than or equal to pie.
Just so you know, given
$\displaystyle x=a\cos(n\theta)$
and $\displaystyle y=a\sin(n\theta)$
You will always get a circle of radius a
$\displaystyle x^2+y^2=a^2\cos^2(n\theta)+a^2\sin^2(n\theta)=a^2$
and if you have
$\displaystyle x=a\cos(n\theta)$
and
$\displaystyle y=b\sin(n\theta)$
You will get an ellipse
$\displaystyle \frac{x^2}{a^2}+\frac{y^2}{b^2}=\frac{a^2\cos^2(n\ theta)}{a^2}+\frac{b^2\sin^2(n\theta)}{b^2}=1$
If the parametric equations were actually $\displaystyle x = \cos t$ and $\displaystyle y = \cos (2t)$ then the cartesian equation would be $\displaystyle y = 2x^2 - 1$ ........
If the parametric equations were actually $\displaystyle x = 2 \sin t$ and $\displaystyle y = \cos (2t)$ then the cartesian equation would be $\displaystyle y = 1 - \frac{x^2}{2}$ ........
Are the equations you posted correct?