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First-principles calculations of electron tunneling transport in nanoscale Ni and Co break-junctions reveal strong dependence of the conductance on the magnetization direction, an effect known as tunneling anisotropic magnetoresistance TAMR. An important aspect of this phenomenon stems from resonant states localized in the electrodes near the junction break. The energy and broadening of these states is strongly affected by the magnetization orientation due to spin-orbit coupling, causing TAMR to be sensitive to bias voltage on a scale of a few millivolts. Our results bear a resemblance to recent experimental data and suggest that TAMR driven by resonant states is a general phenomenon typical for magnetic broken contacts. This effect may be observed in any experiment where a magnetic tip is used to probe electron transport, e.g., spin-polarized scanning-tunneling measurements.