I have trying to prove that this cosine map: $$\frac{r}{4}((a+1)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\ \left(X_n-\frac12\right)-a\right)$$ is a logistic map. What I have done so far: Using various trigonometry identities I have reduced it down to: $$\frac{r(a+1)}{4}(\cos\left(2\arccos\left(\frac{a}{a+1}\right)X_n\right)\cos\left(2\arccos\left(\frac{a}{a+1}\right)\right)-\sin\left(2\arccos\left(\frac{a}{a+1}\right)\right)\sin\left(2\arccos\left(\frac{a}{a+1}\right)\right)) -\frac {ra}{4}$$ This simplifies further but gets very messy. I am not sure how to go further with this. Any help will be greatly appreciated.....
2026-04-02 18:39:54.1775155194
$\cos(2\arccos(\frac{a}{a+1})x$
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