求证(3-sin^4 x-cos^4 x)/2cos^2 x=1+tan^2 x+sin^2 x

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求证(3-sin^4 x-cos^4 x)/2cos^2 x=1+tan^2 x+sin^2 x

求证(3-sin^4 x-cos^4 x)/2cos^2 x=1+tan^2 x+sin^2 x
求证(3-sin^4 x-cos^4 x)/2cos^2 x=1+tan^2 x+sin^2 x

求证(3-sin^4 x-cos^4 x)/2cos^2 x=1+tan^2 x+sin^2 x
证明:
3-sin^4 x-cos^4 x
=1+(1-sin^4 x)+(1-cos^4 x)
=1+(1-sin^2 x)(1+sin^2 x)+(1-cos^2 x)(1+cos^2 x)
=1+(cos^2 x)(1+sin^2)+(sin^2 x)(1+cos^2 x)
=1+cos^2 x+sin^2 x+2(sin^2 x)(cos^2 x)
=2+2(sin^2 x)(cos^2 x)
所以
(3-sin^4 x-cos^4 x)/2cos^2 x
=[2+2(sin^2 x)(cos^2 x)]/2cos^2 x
=sec^2 x+sin^2 x
=1+tan^2 x +sin^2 x