Trig +Nested Radicals = Fun

Geometry Level 4

If \(\sqrt{2 + \sqrt{2+\sqrt2}} \) is in the form of \(A \cos \left( \dfrac BC \pi \right) \), where \(A,B\) and \(C\) are positive integers with \(B,C\) coprime, find the minimum value of \(A+B+C\).

Bonus: Investigate why \(\sqrt { 2+\sqrt { 2+\sqrt { 2 \cdots} } } =2\) using trigonometry.