= Solution
Let $\mathcal K_X^*$ be the sheaf of nonzero meromorphic functions. A local equation for a <divisor on a complex manifold> is determined modulo a nowhere-zero holomorphic factor and hence defines a section of the <divisor sheaf on a complex manifold> $\mathcal K_X^*/\mathcal O_X^*$. Conversely, local representatives of such a section have quotients in $\mathcal O_X^*$, so their zero and pole orders agree on overlaps and define a divisor. The constructions are inverse:
$$
\boxed{\operatorname{Div}(X)\cong H^0(X,\mathcal K_X^*/\mathcal O_X^*).}
$$
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