A polarization dependent first shell analysis of the EXAFS spectra shows that cobalt grows on Cu(100) and Cu(111) in perfect epitaxy : the mean nearest neighbour distance between cobalt atoms parallel to the surface is 2.55 Å (same value as in bulk copper) and the interlayer bond length is 2.50 Å. The contraction of the lattice parameter perpendicular to the surface is in agreement with the continuum elasticity theory. The analysis of further neighbour shells is done with the FEFF code (multiple scattering) modified to take into account the structural anisotropy. Simulations are compared to experiments and show anumbiguously that cobalt has an fcc tetragonal structure on Cu 100, but a distorted hcp structure on Cu(111). The (111) surface of copper is hexagonal. An fcc structure means an ..ABCABC.. stacking, and an hcp structure an ..ABAB.. stacking . The main difference between these two structures is the absence of alignment of the first and fourth neighbours in the hcp structure in grazing incidence . This effect is clear in the calculated and in the experimental Fourier Transforms of the EXAFS spectra.