Given this question is rather long to answer, and I'm losing hope it'll ever be, I just want an answer to this particular claim:
Working on the unitary circle, let $x=1-\cos \theta$ and $t=1-\cos n \theta$. Then how can one produce the following equations:
$1-2z^n+z^{2n}=-2z^nt$
$1-2z+z^{2}=-2zx \text{ ?}$
These arise from setting $z^n = l+\sqrt{l^2-1}$ in
If $l$ and $x$ are the cosines of two arcs $A$ and $B$ of a circle of radius unity, and if the first arc is to the second as the number $n$ is to unity then:
$x = \frac{1}{2}\root n \of {l + \sqrt {{l^2} - 1} } + \frac{1}{2}\frac{1}{{\root n \of {l + \sqrt {{l^2} - 1} } }}$
How can this be proven?