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

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Answer 1 · 2018-04-22T14:21:57

$LHS=cos^4x-sin^4x$

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

$=1*cos2x=cos2x=RHS$

Answer 2 · 2018-04-22T14:23:33

See below

We use the following identities

$a^(2n)-b^(2n)=(a^n+b^n)(a^n-b^n)$

$sin^2x+cos^2x=1$

$cos(a+b)=cosacosb-sinasinb$

Proof

$cos^4x-sin^4x=(cos^2x+sin^2x)(cos^2-sin^2x)=cos^2x-sin^2x=cosxcosx-sinxsinx=cos(x+x)=cos2x$

$square$

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