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Question
Prove that: \[\sqrt{2 + \sqrt{2 + 2 \cos 4x}} = 2 \text{ cos } x\]
Solution
\[LHS = \sqrt{2 + \sqrt{2 + 2\cos4x}}\]
\[ = \sqrt{2 + \sqrt{2\left( 1 + \cos4x \right)}} \]
\[ = \sqrt{2 + \sqrt{2 \times 2 \cos^2 2x}} \left( \because 2 \cos^2 2x = 1 + \cos4x \right)\]
\[ = \sqrt{2 + 2\cos2x}\]
\[= \sqrt{2\left( 1 + \cos2x \right)}\]
\[ = \sqrt{2 . 2 \cos^2 x} \left ( \because 2 \cos^2 x = 1 + \cos2x \right)\]
\[ = 2\text{ cos } x = RHS\]
\[\text{ Hence proved } .\]
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