 Solution

Choose a frame of a line bundle on each $U_i$. On $U_i\cap U_j$, the frames differ by $g_{ij}\in {\mathcal{O}}_X^{\ast }(U_i\cap U_j)$, and compatibility on triple intersections is the Čech cocycle condition $g_{ij}{g}_{jk}=g_{ik}$. Replacing the local frames by units $h_i$ changes $g_{ij}$ by the Čech coboundary $h_i^{-1}{h}_j$. Conversely, a multiplicative one-cocycle glues the trivial line bundles ${\mathcal{O}}_{U_i}$ into a line bundle. Tensor product multiplies cocycles, so

Solved by gpt-5.6-sol high.