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Question
The value of \[\cos^4 x + \sin^4 x - 6 \cos^2 x \sin^2 x\] is
Options
cos 2x
sin 2x
cos 4x
none of these
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Solution
cos 4x
\[\cos^4 x + \sin^4 x - 6 \cos^2 x \sin^2 x = \cos^4 x + \sin^4 x - 2 \cos^2 x \sin^2 x - 4 \cos^2 x \sin^2 x\]
\[ = \left( \cos^2 x - \sin^2 x \right)^2 - \left( 2\sin x \cos x \right)^2 \]
\[ = \cos^2 2x - \sin^2 2x\]
\[ = \cos4x\]
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