Let us consider half angle formula for equilateral triangle in unit circle with initial angle as $\frac{\pi}{3}$
$2\cos\frac{\pi}{6} = \sqrt{2+2\cos\frac{\pi}{3}} = \sqrt{2+1}$
Further half angle when iterated we will get the results as follows
$2\cos\frac{\pi}{3\cdot2^2} = \sqrt{2+2\cos\frac{\pi}{3\cdot2^1}} = \sqrt{2+\sqrt{2+\sqrt{2+1}}}$
$2\cos\frac{\pi}{3\cdot2^3} = \sqrt{2+\sqrt{2+2\cos\frac{\pi}{3\cdot2^1}}} = \sqrt{2+\sqrt{2+\sqrt{2+1}}}$
The pattern is clear and obvious and converges to 2 when it is infinite nested radical
Let us imagine negative signs in the nested radicals and cosine angles
$\sqrt{3} = 2\cos\frac{\pi}{6}$
$\sqrt{2-\sqrt3} = 2\sin\frac{\pi}{3\cdot2^2} = 2\cos\frac{5\pi}{3\cdot2^2}$
$\sqrt{2-\sqrt{2-\sqrt3}} = 2\sin\frac{5\pi}{3\cdot2^3} = 2\cos\frac{7\pi}{3\cdot2^3}$
$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt3}}} = 2\sin\frac{7\pi}{3\cdot2^4} = 2\cos\frac{17\pi}{3\cdot2^4}$
$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2-\sqrt3}}}} = 2\sin\frac{17\pi}{3\cdot2^5} = 2\cos\frac{31\pi}{3\cdot2^5}$
$\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2-\sqrt3}}}}} = 2\sin\frac{31\pi}{3\cdot2^6} = 2\cos\frac{65\pi}{3\cdot2^6}$
It is interesting to note that, the above pattern is amazing in number pattern in numerator as follows
$5 = 2^2+1$; $7 = 2^3-1$; $17 = 2^4+1$; $31 = 2^6+1$; $65 = 2^2+1$; $127 = 2^7-1$;...
This corresponds to Jacobsthal-Lucas numbers here. For $n$ number of $-$ signs cosine angle can be derived as follows
$$2\cos(\frac{2^{n+1}-(-1)^n}{3\cdot2^{n+1}})\pi$$
$$\lim_{n\to\infty} 2\cos(\frac{2^{n+1}-(-1)^n}{3\cdot2^{n+1}})\pi = 2\cos\frac{\pi}{3}=1 $$
Even though it is very easy to derive the numerator by above method, this sequence of nested radical also expresses the geometric progression in a different way. (I hope if I continue here some of you may curse for long posts. So I'm leaving to work out yourself)
This particular finite to infinite nested radical is peculiar and notorious in the sense that it starts with $\frac{\pi}{3}$ and when extended to infinity ends with $\frac{\pi}{3}$ - (circular)
Questions:
1. Please explain about Jacobsthal-Lucas numbers and their significance
(In previous post I was easily able to represent cyclic infinite or finite nested square roots of 2 respectively as $cin\sqrt2$ or $n\sqrt2$ as they involved only number 2. But here we get another number ($\sqrt3$ in this case) at the right extreme of radicals)
2. How can we name above nested radical simply? Please suggest