Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$, and set $\alpha_n = f^{(n)}(0) = P_n(0)$ for every natural integer $n$. Using the identity $2f^{\prime}(x) = f(x)^2 + 1$, show $2\alpha_1 = \alpha_0^2 + 1$ and
$$\forall n \in \mathbb{N}^{\star}, \quad 2\alpha_{n+1} = \sum_{k=0}^{n} \binom{n}{k} \alpha_k \alpha_{n-k}.$$