I'am try to work with Chern class of the coherent sheaves, in this sense. If I have a vector bundle $E$ of rank $r$ and $L$ a line bundle we have the Chern class property
$$c_{r}(E\otimes L) = \sum_{i = 0}^{r}c_{i}(E)c_{1}(L)^{r-i}.$$
I need a similar result for coherent sheaf.
If I have a coherent sheaf $\mathcal{F}$ over a smooth projective variety X we consider a locally free resolution
$$0 \longrightarrow E_{n} \longrightarrow \cdots \longrightarrow E_{0} \longrightarrow \mathcal{F} \longrightarrow 0.$$
We can define the total Chern class of $\mathcal{F}$ by
$$c(\mathcal{F}) = \displaystyle\Pi_{i = 0}^{n} c(E_{i})^{(-1)^{i}}.$$
My question: Is true that $c_{n}(\mathcal{F}\otimes L) = \sum_{i = 0}^{n}c_{i}(\mathcal{F})c_{1}(L)^{n-i},$ for the coherent sheaf $\mathcal{F}$? If is not there exist the similar relationship?
Thank you !!