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Question $1$:

Is $\frac{1}{\pi}\arccos\left(\frac{{\sqrt{2*\sqrt{2*\sqrt{2}*...n}}}}{2}\right)$ always a rational number when each$*$ is either $+$ or $-$ and $n$ may or may not be infinite?

Question $2$

If its a rational number, then how is it related with the signs($*$) and the value of $n$?

I think both questions can be solved by using half angle formulas for sine and cosine but I am not able to do so.

gammatester
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Arkin
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  • Didn't you rather mean rational multiple of $\pi$? – Lucian Jul 14 '14 at 09:52
  • Sure that you mean $\arccos\left(\frac{{\large \sqrt{2\sqrt{2\sqrt{2}...n}}}}{\pi}\right)$ and not $\frac{1}{\pi}\arccos\left(\sqrt{2\sqrt{2\sqrt{2}...n}}\right)?$ – gammatester Jul 14 '14 at 10:02
  • I did it. But you should clarify two other possible issues: (1) Do the * always denote the same operation or can + and - be mixed? (2) The $\arccos \sqrt{2}, \arccos \sqrt{2+ \sqrt{2}},\dots$ are complex. – gammatester Jul 14 '14 at 10:13
  • Its$\arccos{\frac{\sqrt{2+\sqrt{2}}}{2}}$, $*$ can be mixed – Arkin Jul 14 '14 at 10:23

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