Invertible sheaf isomorphic to $\mathcal{O}(div (s))$ but $\mathcal{O}(1)$ on $\mathbb{P}^1$ not?

Question

Let $s \in \Gamma(\mathcal{O}_{\mathbb{P}^1}(1))$ we now that all such sections is in bijective correspondence to linear polynomials $a_0 + a_1 x$, so we take section $s$ wich corresponding to constant $1$. Divisor of such section is trivial (there is no zeros and poles), and Vakhil 14.2.E say that $\mathcal{L} = \mathcal{O}(div (s))$ for any invertible sheaf $\mathcal{L}$ and it section $s$. So $$\mathcal{O}_{\mathbb{P}^1}(1) = \mathcal{O}_{\mathbb{P}^1}(div(s)) = \mathcal{O}_{\mathbb{P}^1}(\text{trivial divisor}) = \mathcal{O}_{\mathbb{P}^1}$$ but it is definitely wrong. Where is mistake?