© 2016 American Physical Society. We analyze the second-class current decays τ-→π-η(′)ντ in the framework of chiral perturbation theory with resonances. Taking into account π0-η-η′ mixing, the π-η(′) vector form factor is extracted, in a model-independent way, using existing data on the π-π0 one. For the participant scalar form factor, we have considered different parametrizations ordered according to their increasing fulfillment of analyticity and unitarity constraints. We start with a Breit-Wigner parametrization dominated by the a0(980) scalar resonance and after we include its excited state, the a0(1450). We follow by an elastic dispersion relation representation through the Omnès integral. Then, we illustrate a method to derive a closed-form expression for the π-η, π-η′ (and K-K0) scalar form factors in a coupled-channels treatment. Finally, predictions for the branching ratios and spectra are discussed emphasizing the error analysis. An interesting result of this study is that both τ-→π-η(′)ντ decay channels are promising for the soon discovery of second-class currents at Belle-II. We also predict the relevant observables for the partner η3(′) decays, which are extremely suppressed in the Standard Model.