Let \( f : [-\pi, \pi] \to \mathbb{R} \) be a continuous function such that \( f(x)>\frac{f(0)}{2} \) for \( |x|<\delta \), where \( 0<\delta<\pi \). Define \( P_{n,\delta}(x) = (1 + \cos x - \cos \delta)^n \), for \( n = 1, 2, 3, \dots \). Then, which of the following statements is TRUE?