Question

Solve :- $\sqrt{2+\sqrt{2+2\cos 4\theta }}=2\cos \theta$

Answer 1

According to question,
$\sqrt{2+\sqrt{2+2cos4\theta }}=2cos\theta$
squaring on both sides,
=> $2+\sqrt{2+2cos4\theta }=4co{s}^2\theta$
=> $\sqrt{2+2cos4\theta }=4co{s}^2\theta -2$
=> $\sqrt{2+2cos4\theta }=2(2co{s}^2\theta -1)$
=> $\sqrt{2+2cos4\theta }=2\left(cos2\theta \right)$
Again squaring,
=> $2+2cos4\theta =4\left(co{s}^{2}2\theta \right)$
=> $2cos4\theta =4\left(co{s}^{2}2\theta \right)-2$
=> $2cos4\theta =2(2co{s}^{2}2\theta -1)$
=> $2cos4\theta =2\left(cos4\theta \right)$
=> LHS=RHS
Hence proved