We can show that the dual bundle $E^*$ with fibers $(E^*)_x = (E_x)^*$ for all $x \in B$ have transition functions $g_{\alpha\beta} = (f_{\alpha\beta}^T)^{-1} : U_\alpha \cap U_\beta \to GL(r,\mathbb R)$.
However, I can't seem to show that the $g_{\alpha\beta}$ satisfy the cocycle conditions, that is $g_{\alpha\alpha} = \text{Id}_F$ for $x \in U_\alpha$, $g_{\alpha\beta} \circ g_{\beta\alpha} = \text{Id}_F$ for $x \in U_\alpha \cap U_\beta$ and $g_{\alpha\beta} \circ g_{\beta\gamma} \circ g_{\gamma\alpha} = \text{Id}_F$ for $x \in U_\alpha \cap U_\beta \cap U_\gamma$.
How might we compute that the cocycle conditions are satisfied?