Showing Chi squared is independent with another variable

[torquerotates]

So I have X1 and X2 are iid standard normal.

Then I have Y=X1^2+ X2^2

and

Z=X1/(X1^2+x2^2)

I'm supposed to find the distribution of Y and Z and then determine if they are independent.

Clearly Y is chi squared with degrees of freedom 2.

But I have no idea how to find the distribution of Z. I know there is a shortcut without using the jacobian, like I did with the Chi Squares, but I'm not sure how to do it.
I know Y and Z are not independent because with some algebra,

Y=Y*Z^(2)+X2
So Y depends on Z and vice versa, therefore they cannot be independent. Is that true?

 

[mfb]

(I moved this thread to the homework section)

I could be interesting to go to polar coordinates (in the X1-X2-plane).

Just finding an equation where Y and Z appear is not sufficient to show a dependence.
Imagine Y=X1, Z=X2, then Y+Z=X1+X2 but they are clearly independent.