Correlation of two dependent variables

Question

I've solved a problem where two Random Variables are dependent and their Covariance $\text{cov}(X,Y)=0$. So will that make the correlation of $X$ and $Y$ also zero or not ?

Answer 1

regardless whether $X$ and $Y$ are dependent or not, correlation is $0$ because $corr(X,Y) = \frac{cov(X,Y)}{\sigma_X \sigma_Y} = 0$

Answer 2

$X \sim \mathcal{N}(0,1)$

We clearly have that $X$ and $X^2$ are dependent. However, $$cov(X,X^2)=E[X^3]-E[X]E[X^2]=0$$

So $$Corr(X,X^2)=\frac{cov(X,X^2)}{\sqrt{var(X)var(X^2)}}=0$$