Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 115 / 2 / c

Each $U_j$ is isomorphic to $\mathbb{C}$ and is therefore contractible. Every complex line bundle over a contractible paracompact space is trivial, so $L{|}_{U_0}$ and $L{|}_{U_1}$ admit the required trivializations.