Expected angle between bivariate normal vectors

Question

I am trying to find an expression for the expected angle between two correlated gaussian vectors, i.e., find $E[\theta] = E\left[\cos^{-1}\left(\frac{\bf x_1'x_2}{\|\bf x_1\|\|x_2\|}\right)\right]$ where $\bf x_1$ and $\bf x_2$ $\in R^2$, \begin{equation} \left(\begin{array}{c} \bf x_1\\ \bf x_2\end{array}\right) \sim N(\bf 0,\Sigma) \end{equation} and \begin{equation} \bf \Sigma = \left(\begin{array}{cc} \bf \Sigma_{11} &\bf \Sigma_{12}\\ \bf \Sigma_{21} &\bf \Sigma_{22} \end{array} \right) \end{equation} and $\bf \Sigma_{12}=\bf \Sigma_{21}' \neq \bf 0$. Assume all covariance submatrices are known. Any help will be greatly appreciated.