[Abridged] We present the results of new near-IR spectroscopic observations of passive galaxies at z>1.4 in a concentration of BzK-selected galaxies in the COSMOS field. The observations have been conducted with Subaru/MOIRCS, and have resulted in absorption lines and/or continuum detection for 18 out of 34 objects. This allows us to measure spectroscopic redshifts for a sample almost complete to K(AB)=21. COSMOS photometric redshifts are found in fair agreement overall with the spectroscopic redshifts, with a standard deviation of ~0.05; however, ~30% of objects have photometric redshifts systematically underestimated by up to ~25%. We show that these systematic offsets in photometric redshifts can be removed by using these objects as a training set. All galaxies fall in four distinct redshift spikes at z=1.43, 1.53, 1.67 and 1.82, with this latter one including 7 galaxies. SED fits to broad-band fluxes indicate stellar masses in the range of ~4-40x10^10Msun and that star formation was quenched ~1 Gyr before the cosmic epoch at which they are observed. The spectra of several individual galaxies have allowed us to measure their Hdelta_F and Dn4000 indices, which confirms their identification as passive galaxies, as does a composite spectrum resulting from the coaddition of 17 individual spectra. The effective radii of the galaxies have been measured on the HST/ACS F814W image, confirming the coexistence at these redshifts of passive galaxies which are substantially more compact than their local counterparts with others that follow the local size-stellar mass relation. For the galaxy with best S/N spectrum we were able to measure a velocity dispersion of 270+/-105 km/s, indicating that this galaxy lies closely on the virial relation given its stellar mass and effective radius.
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