Prove that `8*cos^2x + cos 4x = 8*cos^4x + 1`

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We have to prove that `8*cos^2x + cos 4x = 8*cos^4x + 1`

Starting with the left hand side:

`8*cos^2x + cos 4x`

=> `8*cos^2x + (cos^2(2x) - sin^2(2x))`

=> `8*cos^2x + (cos^2x - sin^2x)^2 - (2*sin x*cos x)^2`

=> `8*cos^2x + cos^4x + sin^4x - 2*cos^2x*sin^2x - 4*sin^2x*cos^2x`

=> `8*cos^2x + cos^4x +(1 - cos^2)^2 - 6*cos^2x(1 - cos^2x)`

=> `8*cos^2x + cos^4x +1 +cos^4x - 2*cos^2x - 6*cos^2x + 6*cos^4x`

=> `8*cos^4x + 1`

This proves the identity: `8*cos^2x +cos 4x = 8*cos^4x + 1`

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