Answer
The rectangular equation for the given parametric equations is ${{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}=1$.
Work Step by Step
Let us consider the provided parametric equations
$\begin{align}
& x={{\cos }^{3}}t \\
& y={{\sin }^{3}}t \\
\end{align}$
And take cubic roots on both sides of the parametric equations
$\begin{align}
& {{x}^{\frac{1}{3}}}=\cos t \\
& {{y}^{\frac{1}{3}}}=\sin t \\
\end{align}$
Then, square both sides of the equations
$\begin{align}
& {{x}^{\frac{2}{3}}}={{\cos }^{2}}t \\
& {{y}^{\frac{2}{3}}}={{\sin }^{2}}t \\
\end{align}$
Then, add the equations
$\begin{align}
& {{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}={{\cos }^{2}}t+{{\sin }^{2}}t \\
& =1
\end{align}$
Thus, the rectangular equation for the given parametric equations is ${{x}^{\frac{2}{3}}}+{{y}^{\frac{2}{3}}}=1$.
