In the first, top rated, comment on the original post linked below, the author wrote the function $r = \dfrac {\cos(x)} {1−\cos(x)}$. I do not understand how they got to this equation. I get how they solved for $x$ in their first step. Can you please explain how they solved for $r$.
This is the picture of the problem, where we are solving for $r$, given $x = \dfrac {n-2}{2n}$
Original Post
Numbers of circles around a circle
Answer
In the right Triangle, we can write $\cos x $ as
Base divided by the hypotenuse
$$\cos x=\frac{r}{r+1}$$
$$\frac{1}{\cos x}=1+\frac{1}{r}$$
$$\frac{1}{r}=\frac{1}{\cos x}-1$$
$$\frac{1}{r}=\frac{1-\cos x}{\cos x}$$
$$r=\frac{\cos x}{1-\cos x }$$
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