\(3\left(sin^8x-cos^8x\right)+4\left(cos^6x-2sin^6x\right)+6sin^4x\)
\(=3\left(sin^2x-cos^2x\right)\left(sin^4x+cos^4x\right)+4\left(cos^2x-sin^2x\right)\left(cos^4x+sin^4x+cos^2x.sin^2x\right)4sin^6x+6sin^4x\)
\(=\left(cos^2x-sin^2x\right)\left(sin^4x+cos^4x+4sin^2xcos^2x\right)-4sin^6x+6sin^4x\)
\(=3cos^4xsin^2x-3cos^2xsin^4x+cos^6x+6sin^4x-5sin^6x\)
\(=3cos^4xsin^2x-3cos^2xsin^4x+cos^6x+sin^4x+5sin^4x\left(1-sin^2x\right)\)
\(=3cos^4xsin^2x+2sin^4xcos^2x+cos^6x+sin^4x\)
\(=cos^4x\left(3sin^2x+cos^2x\right)+sin^4x\left(2cos^2x+1\right)\)
\(=cos^4x\left(3-2cos^2x\right)+sin^4x\left(3-sin^2x\right)\)
\(=3\left(cos^4x+sin^4x\right)-2\left(cos^6x+sin^6x\right)\)
\(=3\left(cos^4x+sin^4x\right)-2\left(sin^4x+cos^4x-sin^2xcos^2x\right)\)
\(=\left(sin^2x+cos^2x\right)^2=1\)
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