We evaluate the nucleon axial form factor, $G_A(q^2)$, and induced pseudoscalar form factor, $G_p(q^2)$, as well as the pion-nucleon form factor, $G_{pi N N}(q^2)$, in lattice QCD. We also evaluate the corresponding nucleon to $Delta$ transition form factors, $C_5^A(q^2)$ and $C_6^A(q^2)$, and the pion-nucleon-$Delta$ form factor $G_{pi NDelta}(q^2)$. The nucleon form factors are evaluated in the quenched theory and with two degenerate flavors of dynamical Wilson fermions. The nucleon to $Delta$ form factors, besides Wilson fermions, are evaluated using domain wall valence fermions with staggered sea quark configurations for pion masses as low as about 350 MeV. Using these form factors, together with an evaluation of the renormalized quark mass, we investigate the validity of the diagonal and non-diagonal Goldberger-Treiman relations. The ratios $G_{pi NDelta}(q^2)/G_{pi NN}(q^2)$ and $2C_5^A(q^2)/G_A(q^2)$ are constant as a function of the momentum transfer squared and show almost no dependence on the quark mass. We confirm equality of these two ratios consistent with the Goldberger-Treiman relations extracting a mean value of $1.61(2)$.
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