It is shown in terms of a fully relativistic spin-polarized ab initio-type approach that in Pt/Co/Pt trilayers two types of anisotropic magnetoresistance (AMR) have to be distinguished: an in-plane and an out-of-plane AMR. The obtained results, namely the magnetic field dependence as well as the thickness dependence of both AMR types are in very good agreement with a very recent experimental study, in which the in-plane as well as the out-of-plane AMR was reported for this system. The difference between the two types of AMR is visualized in terms of layer-resolved resistivities. In particular, it is confirmed that the anisotropic interface magnetoresistance (AIMR) introduced in the recent publication mainly originates in the vicinity of the Co/Pt interfaces.