Cohomology of a tensor product of sheaves
Solution 1:
You need to make some positivity assumptions on $E$ and $F$, as your conclusion is just not true in general. The only case I know of is Le Poitier's vanishing theorem, which says that if $E \otimes F \otimes \omega_X^{-1}$ is ample on a smooth projective variety $X$, then $H^i(X,E \otimes F)=0$ for $i \geq rs$, where $rk(E)=r$ and $rk(F)=s$. This is satisfied for instance if $\omega_X^{-1}$ is nef, and $E$ and $F$ are both ample on $X$, but even here you need $rs$ to be small relative to the dimension of $X$ if you want to say something meaningful.