So I have narrowed down a possible aproach to this by using the Double Angle and Half Angle Formulas, but whenever I try to apply them I get a wrong answer.
Here is the problem:
If \(a_0 = \sin^2 \left( \frac{\pi}{45} \right)\) and \(a_{n + 1} = 4a_n (1 - a_n)\)
for \(n \ge 0\) find the smallest positive integer \(n\) such that \(a_n = a_0\).
