Question

sin4 x - cos4 x = 1 - 2 cos2x
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Answer 1

LHS = SIN⁴X - COS ⁴X

$( { \sin }^{2} x) ^{2} - (\cos^2x) ^{2} = ( \sin {}^{2}x - \cos^{2}x) ( \sin {}^{2}x + \cos^{2}x) \\since \: ( \sin {}^{2}x + \cos^{2}x) = 1 \: and \: \sin^{2} = 1 - \cos^{2} \\ = ( \sin {}^{2}x - \cos^{2}x)(1) \\ = (1 - \cos^{2}x) - \cos^{2}x = 1 - 2 \cos^{2} x$

So LHS = RHS

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