Prove Independence of a Bivariate Normal Distribution

Two variables $X,Y$ are bivariate normally distributed. We know that $Var(X)=Var(Y)$. Show that the two random variables $X+Y$ and $X-Y$ are independent.

I'm feeling pretty stumped by this question, and I hope you guys can help me out.

My foremost instinct is to somehow prove that $\rho=0$, but I have a lot of doubts and questions toward that aim, for which I can't seem to find answers in my text. Can I assume that $X \sim N(\mu_X,\sigma^2)$ and $Y \sim N(\mu_Y,\sigma^2)$? Should I try to integrate the joint PDF of X and Y to find E(XY)?

I think I should set up the transformations:
$U=X+Y$ and $V=X-Y$
but beyond that, I'm completely stumped.

Solution 1:

\begin{align} cov(X+Y, X-Y) &= Var(X)-Var(Y)+Cov(X,Y)-Cov(X,Y)\\ &= 0 \end{align}

Hence, they are uncorrelated.