Solution:
Using Jacobians to go from the joint pdf of U, T to that of X, Y, we see the latter factors into a function of x times a function of y, so X and Y are independent, and they each have the N(0, 1) distribution. Thus, X and Y are i.i.d. standard Normal r.v.s; this result is called the Box-Muller method for generating Normal r.v.s.

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