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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?

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    What is the cosine of an arc? As in the cosine of the angle it forms? Also is the statement "the first arc to the second..." as in, the angle of one arc is the second multiplied by $n$? If yes, this should be related to Chebyshev polynomials.2012-04-14
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    @Sam Why would there by any doubt what the cosine of an arc is? The statement is saying $l = \cos n\theta$ and $x= \cos \theta$ if I'm not reading things backwards-2012-04-14

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