求证:(2-cos²α)(2+tan²α)=(1+2tan²α)(2-sin²α)

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求证:(2-cos²α)(2+tan²α)=(1+2tan²α)(2-sin²α)

求证:(2-cos²α)(2+tan²α)=(1+2tan²α)(2-sin²α)
求证:(2-cos²α)(2+tan²α)=(1+2tan²α)(2-sin²α)

求证:(2-cos²α)(2+tan²α)=(1+2tan²α)(2-sin²α)
证明:
左边=4+2tan^2α-2cos^2α - cos^2α tan^2α
=2+2tan^2α+2-2cos^2α-sin^2α
=2+2tan^2α+2sin^2α-sin^2α
=2 + 2tan^2α + sin^2α
右边=(1+2tan^2α)(1+cos^2α)
=1+cos^2α+2tan^2α+2tan^2αcos^2α
=1+cos^2α+2tan^2α+2sin^2α
=1+1-sin^2α+2tan^2α+2sin^2α
=2+2tan^2α+sin^2α
左边等于右边原式成立.