Question

prove that -----
$\sqrt{2 + \sqrt{2 + \sqrt{2 \cos \: 4 \: theta}}} = 2cos \: theta \:$
​

Answer 1

ANSWER ⤵️

$cos 2Θ = 2 cos ^2 Θ -1$

$cos 4 Θ = 2 cos ^2 (2 Θ) -1$

$1+cos 4 Θ = 2 cos ^2 2 Θ$

-----------------------eq.1

$cos (2 Θ)= 2 cos ^2 Θ -1$

$1+cos 2Θ = 2 cos ^2 Θ$

------------------------eq.2

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

$\sqrt{2 \sqrt{2(2cos \: 2 \ Θ ) } }$

$\sqrt{2 \sqrt{(1 + \cos \: 2 \: Θ)} }$

$\sqrt{2(2 {cos}^{2} \: Θ)}$

${ \sqrt{4 {cos}^{2} Θ}}$

$:\longrightarrow\pink{2cos \: theta}$

$:\longrightarrow\blue{RHS}$

$\purple{BE \: \: BRÃÎÑLY❥}$

$✝HOPE \: IT \: HELPS✝$