How do you prove cos^4(x) - sin^4(x) = cos(2x)?

cos 4 x−sin 4 x = (cos 2 x+sin 2 x)(cos 2 x−sin 2 x)=cos(2x)

From the Pythagorean identity we know that

cos 2 x+sin 2 x=1

so we can write cos 2 x−sin 2 x=cos(2x)

Now cos(2x)=cos(x+x)

From the angle sum identity we have

cos(2x)=cos(x+x)=cos(x)cos(x)−sin(x)sin(x) cos(2x)

=cos(x+x)=cos 2 x−sin 2 x

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