I don't see how you can use Cauchy's theorem here... unless maybe you mean Cauchy'sresiduetheorem.

But that's not necessary either. It's probably easiest to use Cauchy's integral formula. Draw a line through the unit circle such that each pole is inside one of the resulting half-circles. Let be appropriately-oriented paths corresponding to each half-circle such that . Then by Cauchy's integral formulae we have