Locally constant sheaf on a simply connected space I was reading an article and at some point the writer claims that 
1)A locally constant sheaf on a simply connected topological space is a constant sheaf.
2) $H^{i}(U,\mathcal{F})=0 \hspace{0.1cm}\forall i>1$ where U is a homotopically trivial open set and $\mathcal{F}$ a locally constant sheaf.
How could I prove that?
Thank you for your time.

A more abstract way of reformulating the result is that the category of local systems ( = locally constant sheaves) on $X$ with stalk $M$ ($M$ is a $k$-vector space or a module)  is equivalent to the category of representations $\rho : \pi_1(X) \to GL(M)$. In particular, if $X$ is simply connected then every locally constant sheaf is constant.
This also shows that one could compute everything related to the local system from the representation, as an example if $\mathscr L$ is a local system on $D^*$ (punctured disk) then a local system is equivalent to an element $T \in GL(M)$. The cohomology of $\mathscr L$ is the cohomology of the complex $M \overset{d}{\to} M$ with $d = \text{id} - T$. I am not aware of a formula in the general case but in theory this should be possible, using Cech complex.
I think a good reference for this Galois groups and fundamental groups by Tamas Szamuely. 
