Given $X \sim N(0, 1)$ and $Y = X^2$, show that $\rho_{\tiny X, \tiny Y} = 0$

Question

What I tried was:

$\rho{\tiny X, \tiny Y} = \frac{Cov(X, Y)}{\sigma_x \sigma_y} = \frac{Cov(X, X^2)}{1 * 1} = E[(X - \mu_x)(Y - \mu_y)] = E(XY) = E(X)E(Y) = 0$

But the following question is if $X$ and $Y$ are independent. And as far as I know that is a requirement for $E(XY) = E(X)E(Y)$ to hold. Could someone show me how this is done properly? thanks!