Using the sequence of polynomials $(P_n)$ defined by $f^{(n)}(x) = \frac{P_n(\sin x)}{(\cos x)^{n+1}}$ for $f(x) = \frac{\sin x + 1}{\cos x}$ on $I = ]-\pi/2, \pi/2[$, justify that, for every integer $n \geqslant 1$, the polynomial $P_n$ is monic, of degree $n$ and that its coefficients are natural integers.