Question

prove that √(2+√(2+√(2+2cos8∅)=2cos∅​

Answer 1

Here is the answer mate

Answer 2

SOLUTION

L.H.S of the given equation is:

$= > \sqrt{2 + \sqrt{2 + \sqrt{2 + \sqrt{2cos8 \theta} } } } \\ \\ = > \sqrt{2 + \sqrt{2 + \sqrt{2(1 + cos8 \theta)} } } \\ \\ = > \sqrt{2 + \sqrt{2 + \sqrt{2(1 + 2cos {}^{2}4 \theta - 1) } } } \: \: \: \: \: \: \: \: \: [Since \: cos2A = 2cos {}^{2} A - 1]\\ \\ = > \sqrt{2 + \sqrt{2 + \sqrt{2.2cos {}^{2} 4 \theta} } } \\ \\ = > \sqrt{2 + \sqrt{2 + 2cos4 \theta} } \\ \\ = > \sqrt{2 + \sqrt{2(1 + cos4 \theta)} } \\ \\ = > \sqrt{2 + \sqrt{2(1 + 2cos {}^{2}2 \theta - 1) } } \\ \\ = > \sqrt{2 + \sqrt{2.2cos {}^{2}2 \theta } } \\ \\ = > \sqrt{2 + 2cos2 \theta} \\ \\ = > \sqrt{2(1 + cos2 \theta)} \\ = > \sqrt{2(1 + 2cos {}^{2} \theta - 1)} \\ = > \sqrt{2.2 {cos}^{2} \theta} = 2cos \theta$

R.H.S

Hope it helps ☺️