求证：tan^2+cot^2=[2(3+cos4)]/(1-cos4x)

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求证：tan^2+cot^2=[2(3+cos4)]/(1-cos4x)
求证：tan^2+cot^2=[2(3+cos4)]/(1-cos4x)

求证：tan^2+cot^2=[2(3+cos4)]/(1-cos4x)
tan^2x+cot^2x
=sin^2x/cos^2x+cos^2x/sin^2x
=(sin^4x+cos^4x)/(cos^2xsin^2x)
=(sin^4x+cos^4x+2cos^2xsin^2x-2cos^2xsin^2x)/(cos^2xsin^2x)
=[(sin^2x+cos^2x)^2-2cos^2xsin^2x]/(cos^2xsin^2x)
=[1-2cos^2xsin^2x]/(cos^2xsin^2x)
=[1-2cos^2xsin^2x]/(cosxsinx)^2
=[1-2cos^2xsin^2x]/(1/2*sin2x)^2
=4[1-2cos^2xsin^2x]/(sin2x)^2
=4[1-2cos^2xsin^2x]/[(1-cos4x)/2]
=8[1-2cos^2xsin^2x]/(1-cos4x)
=4[2-4cos^2xsin^2x]/(1-cos4x)
=4[2-(sin2x)^2]/(1-cos4x)
=4[2-(1-cos4x)/2]/(1-cos4x)
=4[2-1/2+cos4x)/2]/(1-cos4x)
=4[3/2+cos4x/2]/(1-cos4x)
=2(3+cos4x)/(1-cos4x)

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