Let $X$ be a smooth, projective variety and $F$ a coherent sheaf on $X$. Let $\{U_i\}_{i \in I}$ be an open affine covering of $X$ and $\{f_{ij}\}_{i<j}$ with $f_{ij} \in \Gamma(U_{ij},F)$ satisfying the cocycle condition. Denote by $\alpha \in H^1(X,F)$ corresponding to the collection $\{f_{ij}\}_{i<j}$.

Choose $i_0 \in I$ and a proper affine open subset $V \subset U_{i_0}$. Consider the new open affine covering $\mathcal{V}$ consisting of $V$ and $U_i$ for $i \in I$. We know that $H^1(X,F)$ does not depend on the choice of the covering. What is the Cech cocyle corresponding to $\alpha$ under the new open covering $\mathcal{V}$?

Any hint/reference will be most welcome.