I'm following the book of Okonek, Schneider and Spindler, Vector Bundles on Complex Projective Spaces, and they say that is a useful exercise try to prove Bott's Formula that calculates the cohomology of exterior powers of the cotangent bundle on a projective space, by using induction.

In order to do it, assuming the statement true for for the case p-1 and we have the long exact sequence of cohomologies:

$$0 \to H^{0}(\Omega^{p}_{\mathbb{P^{n}}}) \to C^{n+1}_{p}H^{0}(\mathbb{P^{n}}) \to H^{0}(\Omega^{p-1}_{\mathbb{P^{n}}}) \to H^{1}(\Omega^{p}_{\mathbb{P^{n}}}) \to 0$$ where $C^{n+1}_{p}$ means combination $n+1$, p to p. Here, if $H^{1}(\Omega^{p}_{\mathbb{P^{n}}})$ was zero it will be possible do this because we know the dimension of the other 2 spaces an so we can calculate $\dim H^{0}(\Omega^{p}_{\mathbb{P^{n}}})$. But I don't know how to see it, anyone has a suggestion in order to do it?

Thank you so much in any advance!