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 that is almost complete to KAB = 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 ∼ 4–40×1010M⊙ 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 HδF indices and the strengths of the 4000 Å break, 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 COSMOS HST/ACS iF814W-band 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 effective radius-stellar mass relation. For the galaxy with best S/N spectrum we were able to measure a velocity dispersion of 270±105 km s-1 (error bar including systematic errors), indicating that this galaxy lies closely on the virial relation given its stellar mass and effective radius.