Help please with the problem

#1
W is a standard exponential distribution.
Q is distributed uniformly on [0,2pi]
W and Q are independent.
R=sqrt(2W)
-pi/2<=a<=pi/2 - constant.

U=Rcos(Q)
V=Rsin(Q+a)
Need to prove (U,V) has bivariate normal distribution with correlation equal to sin(a)

Its hard for me to solve it, I tried to find joint distribution of (U,V), but to find this you need to integrate the joint distribution of (W,Q) on complex regions, it is practically impossible. Also I tried to find m.g.f and c.f. of vector (U,V) to show it is Gaussian, but I cant take needed integrals too.

Have someone idea how to solve this?
 

ledzep

Point Mass at Zero
#2
I think using MGFs is the way out here. May be you can get away with integration by adjusting the expressions in integrands so that it will be 1.