This work is devoted to the nonlinear inverse problem of identifying the reaction coefficient in an elliptic boundary value problem from single Cauchy data on a part of the boundary. We then examine simultaneously two elliptic boundary value problems generated from the available Cauchy data. The output least squares method with the Tikhonov regularization is applied to find approximations of the sought coefficient. We discretize the PDEs with piecewise linear finite elements. The stability and convergence of this technique are then established. A numerical experiment is presented to illustrate our theoretical findings.