  We develop a novel self-consistent approach for studying the angle resolved
photoemission spectra (ARPES) of a hole in the t-J-Holstein model giving
perfect agreement with numerically exact Diagrammatic Monte Carlo data at zero
temperature for all regimes of electron-phonon coupling. Generalizing the
approach to finite temperatures we find that the anomalous temperature
dependence of the ARPES in undoped cuprates is explained by cooperative
interplay of coupling of the hole to magnetic fluctuations and strong
electron-phonon interaction.
