prove independence - std normal bivariate

Question.

Suppose that the joint distribution of X and Y is standard bivariate normal with correlation .

Let and .

Show that U and V are uncorrelated.

Answer.

I apply the covariance formula to U and V hoping that it will result in zero.

Since X=U, then E(U)=E(X)=0 and E(X^2)=Var(X)=1 (X is standard normal). Also I take non-random components (like rho) outside the E sign.

Since Cov(X,Y)=E(XY)-E(X)E(Y), and since X and Y are standard normal therefore E(XY)=Cov(X,Y)=Corr (X,Y)=rho

Since I am not sure if this is correct, I would ask you to kindly let me know if there are mistakes.