Question

Find the value of
$\sqrt{2+\sqrt{2+\sqrt{2+2\cos 8\theta }}}$
where $\theta ϵ[0,\pi /16]$

• A. $2\cos \theta$
• B. $2\sin \theta$
• C. 0
• D. 1

Answer 1

The correct option is A $2\cos \theta$

We know,

$\cos 2\theta ={\cos }^2\theta -{\sin }^2\theta =2{\cos }^2\theta -1$

$\Rightarrow 2{\cos }^2\theta =1+\cos 2\theta$

Taking LHS,

$\sqrt{2+\sqrt{2+\sqrt{2+2\cos 8\theta }}}$

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

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

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

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

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

$=2\cos \theta =RHS$