Let $b \in \mathbb{R}$ such that $\cos(b) \in ]0,1[$. Show that the sequence of functions $(f_{b,n})_{n\in\mathbb{N}}$ defined by
$$f_{b,n}(t) = P_{\cos(b)}^{\circ n}\left(\cos^2\left(\frac{\pi}{2}t\right)\right)$$
converges uniformly to 1 on every compact subset of $]-\cos(b), \cos(b)[$ and converges uniformly to 0 on every compact subset of $[-1,-\cos(b)[\cup]\cos(b),1]$.