SOLUTION: 84. confirm the identity cos^4x-sin^4x=cos2x

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Question 174478: 84. confirm the identity cos^4x-sin^4x=cos2x
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
cos^4x-sin^4x=cos(2x)... Start with the given equation.


(cos^2(x))^2-(sin^2(x))^2=cos(2x) ... Factor cos^4x into (cos^2(x))^2. Factor sin^4x into (sin^2(x))^2.


((1+cos(2x))/2)^2-((1-cos(2x))/2)^2=cos(2x) ... Use the power reducing identities



((1+cos(2x))/2+(1-cos(2x))/2)((1+cos(2x))/2-(1-cos(2x))/2)=cos(2x) ... Factor the left side using the difference of squares formula



(2/2)((2cos(2x))/2)=cos(2x) ... Combine the fractions and like terms.


cos(2x)=cos(2x) ... Reduce. So this verifies the identity