= Solution
The exact sequence
$$
0\longrightarrow\mathcal O_X^*\longrightarrow\mathcal K_X^*
\longrightarrow\mathcal K_X^*/\mathcal O_X^*\longrightarrow0
$$
induces a <long exact sequence in sheaf cohomology>. Under $H^0(\mathcal K_X^*/\mathcal O_X^*)=\operatorname{Div}(X)$ and $H^1(\mathcal O_X^*)=\operatorname{Pic}(X)$, its <connecting homomorphism> is the <divisor-to-Picard map> $D\mapsto\mathcal O(D)$. Exactness identifies its kernel with divisors of global nonzero <meromorphic functions>, namely <principal divisors>:
$$
\boxed{\ker(\operatorname{Div}(X)\to\operatorname{Pic}(X))=\operatorname{Prin}(X).}
$$
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