= Solution
Since $X$ is an <irreducible variety>, every nonempty open subset is irreducible and the <sheaf of nonzero rational functions on an irreducible variety> has value $k(X)^*$ there. All its restriction maps between nonempty opens are identities, and restriction to the empty set is <surjective>. Thus $\mathcal K_X^*$ is a <flabby sheaf>, and the introductory argument gives $H^1(X,\mathcal K_X^*)=0$.
Apply the <long exact sequence in sheaf cohomology> to the <short exact sequence of sheaves>
$$
1\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*
\longrightarrow\mathcal D\longrightarrow1.
$$
Its relevant portion is
$$
H^0(X,\mathcal K_X^*)\longrightarrow H^0(X,\mathcal D)
\xrightarrow{\;\partial\;}H^1(X,\mathcal O_X^*)
\longrightarrow H^1(X,\mathcal K_X^*)=0.
$$
The <connecting homomorphism> is therefore <surjective>, with kernel the image of $k(X)^*$. The computation in part (a) now gives
$$
\boxed{\operatorname{Cl}(X)\cong H^1(X,\mathcal O_X^*).}
$$
Concretely, local rational equations $f_i$ of a <divisor on an algebraic curve> differ by regular units on overlaps. Their ratios give the gluing data of an <invertible sheaf>; changing the local equations by units changes those data by a coboundary. This is the transition-function interpretation of the identification with the <Picard group>. Reversing all transition ratios replaces a class by its inverse; either consistent convention gives the same group <isomorphism>.
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