I think this is correct. As for checking the independence, it suffices to show that writes as a product of a function of times a function of (not necessarily probability density functions). For that, you have to expand the exponent and see that the terms vanish. Once you know that the joint density functions writes , you deduce by Fubini that , hence dividing by its integral and by its own, you end up with density functions. But you don't have to compute these integrals to show the independence.

As you can guess, this is a polar change of coordinates. You can show that and .The second part of the problem:

Let the transformation from (X,Y) -> (R,θ) be defined as

find the joint density and marginal densities.

So far on this part I have it solved for X and Y:

I tried plugging in the equations into each other but I don't know how to get rid of the tan(). Any help is appreciated.

For instance, you can write , square this equation and deduce the expression for (using at the end, and ).