Question

Find m : $\sqrt{2+\sqrt{2+\sqrt{2+\cos 8\theta }}}=m\cos \theta$

Answer 1

$\sqrt{2+\sqrt{2+\sqrt{2+2\cos 8\theta }}}=\sqrt{2+\sqrt{2+\sqrt{2(1+\cos 8\theta )}}}$

$=\sqrt{2+\sqrt{2+\sqrt{4{\cos }^{2}4\theta }}}$

$=\sqrt{2+\sqrt{2+2\cos 4\theta }}$

$=\sqrt{2+\sqrt{2(1+\cos 4\theta )}}$

$=\sqrt{2+\sqrt{4{\cos }^{2}2\theta }}$

$=\sqrt{2+2\cos 2\theta }$

$=\sqrt{2(1+\cos 2\theta )}$

$=\sqrt{4{\cos }^2\theta }$

$=2\cos \theta$

$⟹m\cos \theta =2\cos \theta ⟹m=2$