Ok, sort of as a follow up to my previous question, let's recall the de Rham-Weil theorem: Let $F$ be a sheaf on a topological space $X$ and let $\mathcal{L}^{\bullet}$ be an acyclic resolution of $F$. Then $H^{q}(X,\mathcal{F}) \cong H^{q}(\mathcal{L}^{\bullet}({X}))$.

Now let's say I want to compute the cohomology of the complex $\mathcal{L}^{\bullet}({X})$ but my cochain complex

$0\rightarrow \mathcal{F} \rightarrow \mathcal{L}^{1} \rightarrow \mathcal{L}^{2} \rightarrow \mathcal{L}^{3} \cdots $

is not exact, i.e. $\mathcal{L}^{\bullet}$ is not a resolution of $\mathcal{F}$. In my problem (the problem I'm working on) the sheaf $\mathcal{F}$ and all the sheaves $\mathcal{L}^{q}$ are acyclic. How do I go about solving computing the cohomology of $\mathcal{L}^{\bullet}({X})$? I'm a little bit lost here, any tips, ideas or techniques as to how to proceed?