Do you need to use simulation or an analytic approach?
For an analytic approach, you will have to use theorems about product distributions. The distribution of 1 - X where X = U(0,1) still has the same distribution.
The typical way of finding product distributions is to use either theorems regarding general product distributions or to use the characteristic transform which takes a distribution to Fourier space and back again, much in the same way that you take differential equations to fourier or laplace space and back to function space.
Have you come across these techniques before?