We present the results of new near-IR spectroscopic observations of passive galaxies at z \gtrsim 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 \mathit { K } _ { \text { AB } } = 21 . COSMOS photometric redshifts are found in fair agreement overall with the spectroscopic redshifts , with a standard deviation of \sim 0.05 ; however , \sim 30 \% of objects have photometric redshifts systematically underestimated by up to \sim 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 \sim 4 – 40 \times 10 ^ { 10 } M _ { \odot } and that star formation was quenched \sim 1 Gyr before the cosmic epoch at which they are observed . The spectra of several individual galaxies have allowed us to measure their H \delta _ { \text { F } } 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 i _ { F 814 W } -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 \pm 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 .