Why $T_{\alpha}:T^*X\to T^*X$ is a diffeomorphism? With $\alpha$ not necessarily a closed form.

Question

I wanna know why if $\alpha$ a 1-form on $X$, then the next map is a diffeomorphism.

$$T_{\alpha}:T^*X\to T^*X$$ $$\hspace{2.4cm}\beta \mapsto \beta + \alpha_{\pi(\beta)} $$

I know if $\alpha$ is a closed form then the map is a symplectic diffeomorphism, and if $\alpha$ isn't closed it isn't symplectic, but it is a diffeomorphism. I saw some info in this question Coordinate-free proof of non-degeneracy of symplectic form on cotangent bundle but I really don't catch it.