Let $f$ be defined on $I = ]-\pi/2, \pi/2[$ by $f(x) = \frac{\sin x + 1}{\cos x}$. Show that there exists a sequence of polynomials $\left(P_n\right)_{n \in \mathbb{N}}$ with real coefficients such that
$$\forall n \in \mathbb{N}, \forall x \in I, \quad f^{(n)}(x) = \frac{P_n(\sin x)}{(\cos x)^{n+1}}$$
Make explicit the polynomials $P_0, P_1, P_2, P_3$ and, for every natural integer $n$, express $P_{n+1}$ as a function of $P_n$ and $P_n^{\prime}$.