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Question
Prove that: \[1 + \cos^2 2x = 2 \left( \cos^4 x + \sin^4 x \right)\]
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Solution
\[LHS = 1 + \cos^2 2x\]
Using the identity
\[LHS = 1 + \left( \cos^2 x - \sin^2 x \right)^2 \]
\[ = 1 + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x\]
\[= \left( \cos^2 x + \sin^2 x \right)^2 + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x \left[ \because \cos^2 x + \sin^2 x = 1 \right]\]
\[ = \cos^4 x + \sin^4 x + 2 \cos^2 x \sin^2 x + \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x\]
\[ = 2( \cos^4 x + \sin^4 x) = RHS\]
\[\text{ Hence proved } .\]
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