Suppose X and Y are two independent random variables each distributed as $Uniform(0,1)$.

1) Find the joint distribution of X and Y.

2) Let $\displaystyle U = \cos(2 \pi Y)\sqrt{-2\ln(X)}$ and $\displaystyle $V = \sin(2\pi Y)\sqrt{-2\ln(X)}$$. Find the joint distribution

of U and V assuming that the transformation is one-to-one?

3) Find the marginal distributions of U and V?

For 1, I got that my distribution is 1 by multiplying the two distributions together. For 2, I have been getting stuck trying to get my equations in terms of X and Y to perform the transformation. I simplified and got $\displaystyle Y = $$\displaystyle (\frac{1}{2})$$ $\displaystyle \tan^{-1}(\frac{V}{U})$$ but something seems wrong about this and am having trouble trying to get X. For 3, I am obviously stuck and even not totally sure on the support. Any help is appreciated.