We calculate the 2 fermion-2 boson and the 4 fermion scattering amplitudes at one loop in the NSR formulation of the heterotic and type II superstring theories using the covariant fermion emission vertex. The results are shown to agree with well known amplitudes calculated in the light-cone gauge in the Green-Schwarz formulation. The agreement of the 4-fermion amplitude entails the use of a new ϑ-function identity which we also prove.