For the differentiation of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, why is the substitution $x = a \cos^3\theta$ legal?

Question

While looking at a solution for finding the derivative of $x^{\frac23} + y^{\frac23} = a^{\frac23}$, the book uses:

Let $x = a \cos^3\theta$ and $y = a\sin^3\theta$

However, why would that substitution be legal? $\cos\theta$ and $\sin\theta$ only range between $-1$ and $1$, which would imply that $x$ is somewhere between $a\cdot (-1)$ and $a\cdot (1)$ but $x$ could be $2a$ or $3a$ or any other value of multiplication by $a$ aswell. Why is the above substitution legal?

Answer 1

It is valid because lets say you want $x$ to be $8a$, then

$(8a)^{\frac 2 3} + y^{\frac 2 3}=a^{\frac 2 3}$

$y^{\frac 2 3}=-3a^{\frac 2 3}$ Has no real solution for $y$ because the left hand side is a square and the right hand side is a negative number.

The idea is the same for the parametrization for the unit circle, $x^2+y^2=1$, as both $x$ and $y$ can only range between $0$ to $1$, so we can let $x=cos\theta$ and $y=sin\theta$