I want to convert the parametrized curve $\gamma(t) = (\cos^{3}(t), \sin^{3}(t)), \ t \in \mathbb{R}$ to a level curve.
Let $C = \{(x,y) \in \mathbb{R^{2}}: x^{2/3} + y^{2/3} = 1\}$. I claim that $x^{2/3} + y^{2/3} = 1$ describes the same points in the plane as $\gamma$.
If we let $x = \cos^{3}(t)$ and $y = \sin^{3}(t)$, then $x^{2/3} + y^{2/3} = 1$ is satisfied so the image of $\gamma \subset C$. However, I'm having trouble showing the reverse inclusion (ie. $C \subset$ image of $\gamma$). I've seen in many places where they claim these two sets to be equal but I don't see how to show it.