Let $R$ and $\Theta$ be zero mean independent Gaussian random variables with variances $\sigma\_R^2$ and $\sigma\_\Theta^2$ respectively. Hence their covariance matrix would be diagonal give by
$$C\_{R,\Theta}=\begin{bmatrix}\sigma\_R^2 & 0 \\ 0 & \sigma\_\Theta^2\end{bmatrix}$$
Now, let's define transformations given by $X = R\cos (\Theta)$ and $Y = R \sin (\Theta)$. 

The question is, what would be $C\_{X,Y}$, the covariance matrix of transformed random variable $X$ and $Y$?

Thank you in advance.