It would be easier to use MGFs.
You have random variables X and Y which are Bivariate Normal. . How do you show that random variables U = X+Y and V = X-Y are bivariate normal with parameters mu1+mu2 and mu1-mu2. I took the Jacobian of the variables to get absolute value of -1/2. I substitued X= (U+V)/2 and Y=(U-V)/2 into the probability density function for the bivariate normal. The simplification of the formula has be baffled. Any assistance would be appreciated.