Show that there exists a real $a > 0$ such that
$$\forall \theta \in [-\pi,\pi], 1-\cos\theta \geq a\theta^2.$$
Deduce that there exist three reals $t_0 > 0$, $\beta > 0$ and $\gamma > 0$ such that, for all $t \in ]0,t_0]$ and all $\theta \in [-\pi,\pi]$,
$$\left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\beta(t^{-3/2}\theta)^2} \quad \text{or} \quad \left|\frac{P(e^{-t}e^{i\theta})}{P(e^{-t})}\right| \leq e^{-\gamma(t^{-3/2}|\theta|)^{2/3}}.$$