prove independence - std normal bivariate

Question.

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

Let $\displaystyle U=X$ and $\displaystyle V=\frac{Y-{\rho}X}{\sqrt{1-\rho^2}}$.

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.

$\displaystyle Cov(U,V)=E(U,V)-E(U)E(V)=E(UV)=E(X\frac{Y-{{\rho}X}}{\sqrt{1-{\rho}^2}})=\frac{1}{\sqrt{1-{\rho}^2}}E(XY-{\rho}X^2)=$

$\displaystyle =\frac{1}{\sqrt{1-{\rho}^2}}(E(XY)-{\rho}E(X^2))=\frac{1}{\sqrt{1-{\rho}^2}}(\rho-\rho)=0 $

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.