How to prove the following curious nested radical identity?

Question

I have found the following nested radical representation.

By using the triple angle formula for the cosine, $\cos 3\theta$, and making $\theta = 1^\circ$, we get the cubic equation $ 4x^3-3x = \cos 3^\circ $. In this step , I'll use a trick that is rarely used, by expressing $ x $ as

$ x = \frac{1}{2}\sqrt{3+\frac{\cos 3^\circ}{x}} $

Now I just iterate for x and in this way I get that

$ \cos 1^\circ= \frac{1}{2}\sqrt{3+\frac{\cos 3^\circ}{\frac{1}{2}\sqrt{3+\frac{\cos 3^\circ}{\frac{1}{2}\sqrt{3+\frac{\cos 3^\circ}{\frac{1}{2}\sqrt{3+...}}}}}}} $

Numerical calculation shows that this periodic radical converges to $ \cos 1^\circ$ We have that $\cos 1^\circ \approx 0.9998476$

In the following link, the nested radical is evalued

http://m.wolframalpha.com/input/?i=1%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%2F%281%2F2%283%2Bcos%28pi%2F60%29%29%5E1%2F2%29%29%5E1%2F2%29%29%5E1%2F2%29%29%5E1%2F2%29%29%5E1%2F2%29%29%5E1%2F2%29%29%5E1%2F2+to+20+digits&x=0&y=0

How to prove rigorously that the left hand side is the limit of the right hand side? That's to say, I need a proof that the nested radical converges to $ \cos 1^\circ $

Answer 1

Use the Banach fixed-point theorem.

Let $T(x)=\frac{1}{2}\sqrt{3+\frac{\cos 3^\circ}{x}}$. It suffices to prove that there exists a constant $c$ ($=\frac{\sqrt{3}}{2}$) such that $|T(x)-T(y)|<c|x-y|$ for any $x,y\geq\frac{\sqrt{3}}{2}$.

For the second step below, use the fact that $|\sqrt{a}-\sqrt{b}|<|a-b|$ when $a,b>1$. $\displaystyle \left|T(x)-T(y)\right|=c\left|\sqrt{1+\frac{\cos 3^\circ}{3x}}-\sqrt{1+\frac{\cos 3^{\circ}}{3x}}\right|\leq\left|\frac{\cos 3^\circ}{3x}-\frac{\cos 3^\circ}{3y}\right| <\left|\frac{1}{x}-\frac{1}{y}\right|=\frac{|y-x|}{xy}<|x-y|$