= Solution
{c}
The <sheaf of nonzero rational functions on an irreducible variety> $\mathcal K_X^*$ is a <flasque sheaf>. Indeed, its sections on any nonempty open subset are $K^*$: irreducibility makes nonempty open subsets connected and makes any two of them meet, so locally constant rational-function values must agree. Restrictions between such opens are identities; restriction to the empty set is also surjective. The preliminary argument therefore gives
$$
H^i(X,\mathcal K_X^*)=0\qquad(i>0).
$$
Apply the <long exact sequence in sheaf cohomology> to the <short exact sequence of sheaves> of multiplicative <abelian groups>
$$
1\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*\longrightarrow\mathcal Q\longrightarrow1.
$$
Its relevant part is
$$
K^*\longrightarrow\Gamma(X,\mathcal Q)\longrightarrow H^1(X,\mathcal O_X^*)\longrightarrow H^1(X,\mathcal K_X^*)=0.
$$
Part (b) identifies the first map with the <principal divisor> map $K^*\to\operatorname{Div}(X)$. Exactness therefore identifies its cokernel with $H^1(X,\mathcal O_X^*)$. Combining this with part (a) gives
$$
\boxed{\operatorname{Cl}(X)\cong\operatorname{Pic}(X)\cong H^1(X,\mathcal O_X^*).}
$$
The connecting map sends a divisor to the class of the unit ratios of its local rational representatives; these are precisely the transition data of the associated <line bundle>, up to the usual choice of inverse transition convention.
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