How do you prove $cos^4(x)-sin^4(x)=cos2x$ ?

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Answer 1 · 2015-04-19T05:11:12

$cos^4(x) - sin^4(x) = cos(2x)$

Remember the double angle formula for cosine:
$color(blue)(cos(2x)=cos^2(x)-sin^2(x)$

Plugging it into the right hand side:
$cos^4(x) - sin^4(x) = cos^2(x)-sin^2(x)$

Using differences of squares on the left side:
$(cos^2(x) + sin^2(x)) (cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)$

And since $color(blue)(cos^2(x) + sin^2(x)=1$
$1(cos^2(x) - sin^2(x)) = cos^2(x)-sin^2(x)$

How do you prove cos^4(x)-sin^4(x)=cos2xcos^4(x)-sin^4(x)=cos2x?