The matter content of jets in active galactic nuclei is examined in a new way. We model the dynamical expansion of its cocoon embedded in the intra-cluster medium (ICM). By comparing the observed shape of the cocoon with that expected from the theoretical model, we estimate the total pressure (\(P_{\rm c}\)) and electron temperature (\(T_{e}\)) of the cocoon. The number density of the total electrons (\(n_{e^{-}}\)) is constrained by using the non-thermal spectrum of the hot spot and the analysis of the momentum balance between the jet thrust and the rum pressure of ICM. Together with the obtained \(P_{\rm c}\), \(T_{e}\) and \(n_{e^{-}}\), we constrain the matter content in the jets. We find that, in the case of Cygnus A, the ratio of number density of protons to that of electrons is of order of \(10^{-3}\) . This implies the existence of a large number of positron in the jet.