Let $\Delta_{\mathbb{P}^1}$ be the diagonal line of $\mathbb{P}^1 \times \mathbb{P}^1$, and $\mathcal{N}_{\Delta_{\mathbb{P}^1}}$ be its normal sheaf. I heard that $\mathcal{N}_{\Delta_{\mathbb{P}^1}} \simeq T\mathbb{P}^1 \simeq \mathcal{O}_{\mathbb{P}^1}(2)$.
My questions are
1. How can I prove this equality?
2. Can I think this normal sheaf as
$(\mathcal{O}_{\mathbb{P}^1}(1) \boxtimes
\mathcal{O}_{\mathbb{P}^1}(1))|_{\Delta_{\mathbb{P}^1}}$ ?
Thanks in advance.