Question Number 18841 by mondodotto@gmail.com last updated on 30/Jul/17
Answered by Tinkutara last updated on 31/Jul/17
$$\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{8}\theta\right)}}} \\ $$$$=\:\sqrt{\mathrm{2}\:+\:\sqrt{\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{4}\theta\right)}}\:=\:\sqrt{\mathrm{2}\left(\mathrm{1}\:+\:\mathrm{cos}\:\mathrm{2}\theta\right)} \\ $$$$=\:\mathrm{2}\:\mathrm{cos}\:\theta \\ $$