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#