Find all roots of $8x(2x^2-1)(8x^4-8x^2+1)=1$ such that $0<x<1$

Question

Find all roots of the equation $$8x(2x^2-1)(8x^4-8x^2+1)=1$$ such that $0 < x < 1$.

Answer 1

Hint: Put $x=\cos \alpha$, for some $\alpha\in (0,{\pi\over 2})$.

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Then you get $$8 \cos \alpha \cos 2\alpha \cos 4\alpha =1$$

If we multiply this with $\sin \alpha$ we get:

$$ 8\sin \alpha \cos \alpha \cos 2\alpha \cos 4\alpha =\sin \alpha$$ so $$ 4\sin 2\alpha \cos 2\alpha \cos 4\alpha =\sin \alpha$$ so $$ 2\sin 4\alpha \cos 4\alpha =\sin \alpha$$ so $$ \sin 8\alpha =\sin \alpha$$ so ...