I suggest that you look at what happens in the universal cover given by the Poincare disk model. The five geodesics are quite easy to see. In the picture below, the greyed zone is a fundamental domain for the pair of pants. The five sought geodesics are in bold. The book of Peter Buser, *geometry and spectra of hyperbolic surfaces*, is a good reference concerning the representation of surfaces as quotients of the Poincare disk.

[![five-geodesics-in-poincare-disk][1]][1]

  [1]: https://i.sstatic.net/zcqlj.png