Cech and Sheaf (derived-functor) cohomologies are isomorphic on a paracompact space $X$ with the sheaf being, for example, $\underline{\mathbb{C}}^*_M$, the sheaf of $\mathbb{C}^*$-valued functions on $X$.

The proof uses partititions of unity along with hypercohomology and results from spectral sequences. If this ISN'T a black box theorem, I'd love a concise explanation :)