Let $$ \cdots\rightarrow \mathcal{E}^{j-1}\xrightarrow{d_{j-1}}\mathcal{E}^{j}\xrightarrow{d_{j}}\mathcal{E}^{j+1}\rightarrow\cdots $$ be a complex of holomorphic vector bundles over a manifold $X$, and let $\tilde{\mathcal{E}}^{\bullet}$ be another such complex.
A quasi-isomorphism is a co-chain map $f^{\bullet}:\mathcal{E}^{\bullet}\to\tilde{\mathcal{E}}^{\bullet}$ such that the induced maps $f^{p}_{*}:H^{p}(\mathcal{E}^{\bullet})\to H^{p}(\tilde{\mathcal{E}}^{\bullet})$ are all isomorphisms.
I am trying to show that if $X=\mathbb{C}^{n}$ and all the bundles are trivial, then a quasi-isomorphism is (cochain homotopic to), an isomorphism of cochain complexes.
I'm not quite sure how to proceed, but I will detail my thoughts so far. I want to take the inverse on cohomology, $(f^{p}_{*})^{-1}:H^{p}(\tilde{\mathcal{E}}^{\bullet})\to H^{p}(\mathcal{E}^{\bullet})$, and find a cochain map $\tilde{f}^{\bullet}:\tilde{\mathcal{E}^{\bullet}}\to\mathcal{E}^{\bullet}$, which induces this inverse, i.e. a map for which $\tilde{f}^{p}_{*}=(f^{p}_{*})^{-1}$. The next step would then be to show that $f^{\bullet}$ is cochain-homotopic to the inverse of $\tilde{f}^{\bullet}$, i.e.: $$ f^{p}-(\tilde{f}^{p})^{-1}=h^{p+1}d_{i}+\tilde{d}_{i-1}h_{i} $$ for some $h^{i}:\mathcal{E}^{i}\to\tilde{\mathcal{E}}^{i-1}$.
To do so, I will have to somehow use the fact that these vector bundles are trivial, but I am not sure how this will come up. Perhaps this places some useful constraint on the cohomologies. I feel like this should be a lot easier than I have made it, and any help would be much appreciated.