Question

Prove that:-
$2 \cos a = \sqrt{2 \sqrt{2 \sqrt{2 + 2 \cos8a } } }$

Answer 1

$2 \cos(a )= \sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos(8a) } } } \\ rhs \\ = \sqrt{2 + \sqrt{2 + \sqrt{2 + 2 \cos(8a) } } } \\ = \sqrt{2 + \sqrt{2 + \sqrt{2 + 2(2 { \cos(4a) }^{2} - 1)} } } \\ = \sqrt{2 + \sqrt{2 + \sqrt{2 + 4 { \cos(4a) }^{2} - 2 } } } \\ = \sqrt{2 + \sqrt{2 + \sqrt{4 { \cos(4a) }^{2} } } } \\ = \sqrt{2 + \sqrt{2 + 2 \cos(4a) } } \\ = \sqrt{2 + \sqrt{2 + 4 { \cos(2a) }^{2} - 2} } \\ = \sqrt{2 + \sqrt{4 { \cos(2a) }^{2} } } \\ = \sqrt{2 + 2 \cos(2a) } \\ = \sqrt{2 + 4 { \cos(a) }^{2} - 2} \\ = \sqrt{4 { \cos(a) }^{2} } \\ = 2 \cos(a) \\ lhs$