  The atomic environments involved in the magnetostriction effect in
CoFe$_2$O$_4$ and La$_{0.5}$Sr$_{0.5}$CoO$_3$ polycrystalline samples have been
identified by differential extended x-ray absorption fine structure (DiffEXAFS)
spectroscopy. We demonstrate that cobalt atoms at octahedral sites are
responsible for their magnetostriction. The analysis of DiffEXAFS data
indicates that the local-site magnetostrictive strains of Co atoms are reversed
in these two oxides, in agreement with the macroscopic magnetostriction. For
the CoFe$_2$O$_4$ spinel, a large negative strain along the (100) direction has
been determined for the CoO$_6$ octahedron causing a tetragonal contraction in
contrast with the La$_{0.5}$Sr$_{0.5}$CoO$_3$ perovskite, where a positive
moderate strain along the (100) direction was found resulting in a tetragonal
expansion. The different local-site magnetostriction is understood in terms of
the different valence and spin state of the Co atoms for the two oxides. The
macroscopicmagnetostriction would be explained then by the relative change in
volume, either contraction in CoFe$_2$O$_4$ or expansion in
La$_{0.5}$Sr$_{0.5}$CoO$_3$, when the tetragonal axis of the Co site is
reoriented under an externally applied magnetic field.
