In spot-scanning proton therapy, highly precise beam control is required in the treatment nozzle such that the proton beam does not spread out during transportation by restraining the divergence of the beam angle and spot size, simultaneously. In order to evaluate the beam-broadening behaviour induced by passing through the various nozzle components, we have developed a new method to calculate the angular divergence profile of a proton beam in the nozzle. The angular divergence of the proton beam for each nozzle component is calculated by the Monte Carlo simulation code, Geant4, assuming that the initial beam has no divergence. The angular divergence profiles generated in the various nozzle components are then fitted by the analytic function formula with triple Gaussian distributions. The fitted profiles can be treated like analytic response functions and the angular divergence profile in the nozzle can be easily and systematically calculated by using a convolution theorem. The beam-broadening behaviour during transportation in the nozzle is carefully evaluated. The beam profiles are well-characterized by the proposed angular divergence analysis, i.e. triple Gaussian profile analysis. The primary Gaussian part of the beam profile is mainly generated by air and dose monitors with plate electrode components. The secondary and tertiary Gaussian parts are so-called wide-angle scattering and generated mainly by spot-position and profile monitors with metal window and wire components. The scattering of the nozzle component can be analysed using the proposed response function method for the angular distribution. Multiple convolved angular scattering can be determined from the response function of the individual nozzle components. The angular distribution from small to large angle regions can then be quantitatively evaluated by the proposed method. The method is quite simple and generalized, and is a straightforward way to understand the nozzle and component characteristics related to the beam-broadening behaviour.