KöMaL 2016 B4801

Define the sequence $f_n$ of functions by the following recurrence relation: $f_0(x) = f_1(x) = 1, \mathrm{~and ~for ~} n\ge 2 f_n(x) = f_{n-1}(x) \cdot 2\cos(2x) - f_{n-2}(x).$ Determine the number of roots of $f_n(x)$ in the interval $[0,\pi]$.