On Well's book, lemma 4.4 says there is a one-to-one correspondence between the equivalence classes of holomorphic line bundles on $X$ and the elements of the cohomology group $H_1(X,\mathcal{O}^*)$, where $\mathcal{O}^*$ means non-vanishing holomorphic structure sheaf.
Now I wonder if there is a one-to-one correspondence between the equivalence classes of differentiable line bundles on $X$ and the elements of the cohomology group $H_1(X,\epsilon^*)$, where $\epsilon^*$ means non-vanishing differentiable structure sheaf.
The proof of the first statement is here:
I think the second statement can be proved by mimicking the proof above, am I correct?