We present measurements of the anisotropic magnetoresistance (AMR) of La_0.75Sr_0.25MnO₃ films deposited on (001) SrTiO₃ substrates, and a model that describes the experimental results. The model, based on the electronic structure of manganites plus the spin-orbit coupling, correctly accounts for the dependence of the AMR on the direction of the current to the crystalline axes. We measure an AMR of the order of 10^-3 for the current I parallel to the [100] axis of the crystal and vanishing AMR for I||[110], in agreement with the model predictions. Further, we calculate the planar Hall effect and show its connection to AMR.