Finite field/Smooth projective curve/Vector bundel/Etale trivializable and Frobenius periodicity/Fact

Let ${}K$ denote a finite field (or the algebraic closure of a finite field) and let ${}X$ be a smooth projective curve over ${}K$. Let ${}{\mathcal {S}}$ be a locally free sheaf over ${}X$.

Then ${}{\mathcal {S}}$ is étale trivializable if and only if there exists some ${}n$ such that ${}F^{n*}{\mathcal {S}}\cong {\mathcal {S}}$.

Write a proof