I am looking for classical results of Algebraic Geometry that can be proved using cohomology. For example, Riemann-Roch Theorem and Bezout Theorem admits short proofs (providing that you know enough about cohomology) using this kind of techniques. Are there any others? Which is your favourite one?
What about Pappus Theorem or any other projective geometry theorem?
I guess the Weil conjectires are one of the best examples given the key role they played as motivation for introducing many cohomology theories, most famously étale cohomology.