I need to prove that for $\pi: E → M$ a vector bundle then exists a canonical symplectomorphism $T^{*}E^{*}\cong T^{*}E$. I star with definitions: Let be $\pi : E → M$ a vector bundle, $\pi$ is a continuous surjection, for every $x\in M$ then $\pi^{-1}(\{x\})$ has the vectorial structure and it is finite dimensional and such that: every point in $M$ there is a neighborhood $U$, there is $ n$ natural number and some diffeomorphism
${\displaystyle \varphi \colon U\times {\mathbb {R} }^{n}\to \pi ^{-1}(U)}$ such that for all $x\in U$
- $\pi φ(x, v) = x$ for all $v\in R^{n}$
- the app $ v\rightarrow φ(x, v)$ is a ismomorphism between $ R^{n}$ and $π^{-1}(\{x\})$.
The other side we have that a symplectomorphism is a diffeomorphism between two symplectic manifolds $(M,\omega)$ and $(N,\omega')$ such that
$f^{*}\omega '=\omega$ where $f^{*}$ is the pullback of $f$
Besides, $E^{*}$ is the dual of $E$ and $T^{*}E^{*}$ represent the pullback bundle.
Now i guess that i need to show that exists some function $g: (T^{*}E^{*},\omega_1)\to (T^{*}E,\omega_2)$ such that $g^{*}\omega_2=\omega_1$, so how can to start or how i need to define $g$?
$E^{*}$ and $T^{*}E^{*}$ are simplectic manifolds and i guess that $T^{*}:E^{*}\to E$ such that $T^{*}E^{*}=\{(h,e)\in E^{*}\times E : g(h)=\pi(e)\}$ am I right? This excersise is very difficult to me, i dont know how to start, please can give me hints?
Thank you.
Note: I am not sure about my definitions about $T^{*}E^{*}$ just it was intuition but i will appreciate if somebody can help me understand this point too.
The other side i found some recipe and i attach next, where an 1-form tautological is $\alpha= \sum \xi_i dx_i$ with $x_1,\ldots,x_n$ are coordinates in $U\subset E$ and $x_1,\ldots,x_n,\xi_1,\ldots,\xi_n$ are coordinates of $T^{*}E$ for example and the canonical 2-form is given by $$\omega=\sum d\xi_i\wedge dx_i$$ The book is Lectures on symplectic geometric , Ana Cannas Da Silva. Even this i am stuck i do not how to proceed. 
As a @Chris H said before, when the bundle is trivial we have the following:
$T^{*}E$ is isomorphism with $$T^{*}M \times \mathbb{R^{n}}\times \mathbb {R^{n^{*}}}$$ and
$T^{*}E^{*}$ is isomorphism with $$T^{*}M \times \mathbb{R^{n^{*}}}\times \mathbb{R^{n}}$$
then the canonical forms in $T^{*}E$ and $T^{*}E^{*}$ can be describe by $$\sum^{n}dx_i\wedge dy_i +dr_i\wedge ds_i$$ and $$\sum^{n}dx_i\wedge dy_i +ds_i\wedge dr_i$$. Therefore, we can define a difeomorphism by $$\phi:T^{*}E\to T^{*}E^{*} $$ by $\phi(x,y,s,r)=(x,y,r,s)$ and this satisfies that $\phi^{*}\omega_1=\omega_0$.
How can continued by the general case?
Thank you