Let $\pi:P\to M$ a smooth $G$-principal bundle with action $\beta:G\times P\to P$, $G$ a Lie group and $M$ a diff. manifold. Let also $T^*P$ be its cotangent bundle and $\sigma$ be a section of this latter. I was wondering if a change of trivialization of the cotangent bundle may be seen as induced by a change of trivialization of $P$ and thus as given by the action of $\beta_g$ with $g\in G$, namely as the action over $P$ and the pull-back of the covector, like: $$(p,\sigma_p)\to(\beta_g(p),\beta^*_g\sigma_p).$$
Does it make sense?