= Solution
Put $\mathcal D=\mathcal K_X^*/\mathcal O_X^*$, the quotient <sheaf of abelian groups>. At a closed point $P$, the <discrete valuation> gives an <isomorphism>
$$
\mathcal D_P=k(X)^*/\mathcal O_{X,P}^*
\xrightarrow{\;\sim\;}\mathbb Z,\qquad [f]\longmapsto v_P(f).
$$
Indeed, a <local parameter on a smooth algebraic curve> has valuation one, and the valuation kernel is precisely the <unit group> of the <discrete valuation ring>. Thus a local section represented by $f$ determines its local <principal divisor>, and changing $f$ by a <regular function> which is a unit does not change those coefficients.
These local valuations identify $\Gamma(X,\mathcal D)$ with $\operatorname{Div}(X)$. To see that they give finite support, represent a section locally by finitely many <rational functions on an algebraic variety>, using <Noetherian Zariski topology> to take a finite cover. Each representative has only finitely many zeros and poles, so their combined support is finite. Conversely, for $D=\sum n_PP$, choose at each support point a <local parameter> $t_P$ and shrink its neighbourhood to exclude every other zero or pole of $t_P$ and every other support point. The local rational equation $t_P^{n_P}$ represents the desired valuations there; on the complement of the support use $1$. On overlaps these equations differ by units, so their classes glue in the quotient <sheaf>. Finally, a section with every valuation zero has zero germ everywhere and is the identity section. Hence the identification is both <surjective> and <injective>.
The map on <global sections> sends $f\in k(X)^*$ to $\operatorname{div}(f)$. Therefore its <cokernel> is
$$
\boxed{\operatorname{coker}\bigl(H^0(X,\mathcal K_X^*)\to
H^0(X,\mathcal K_X^*/\mathcal O_X^*)\bigr)
=\operatorname{Div}(X)/\operatorname{Prin}(X)
\cong\operatorname{Cl}(X).}
$$
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