若sin^4a/sin^2b+cos^4a/cos^2b=1,证明sin^4b/sin^2a+cos^4b/cos^2a=1.(不用向量怎么证?)

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若sin^4a/sin^2b+cos^4a/cos^2b=1,证明sin^4b/sin^2a+cos^4b/cos^2a=1.(不用向量怎么证?)

若sin^4a/sin^2b+cos^4a/cos^2b=1,证明sin^4b/sin^2a+cos^4b/cos^2a=1.(不用向量怎么证?)
若sin^4a/sin^2b+cos^4a/cos^2b=1,证明sin^4b/sin^2a+cos^4b/cos^2a=1.(不用向量怎么证?)

若sin^4a/sin^2b+cos^4a/cos^2b=1,证明sin^4b/sin^2a+cos^4b/cos^2a=1.(不用向量怎么证?)
证明:
输入过于麻烦,用换元法吧
设A=sin²A,B=sin²B
∵ sin^4a/sin^2b+cos^4a/cos^2b=1
即A²/B+(1-A)²/(1-B)=1
∴ A²(1-B)+(1-A)²B=B(1-B)
∴ A²-A²B+B-2AB+A²B=B-B²
∴ A²-2AB=-B²
∴ A²-2AB+B²=0
∴ (A-B)²=0
∴ A=B
∴ sin^4b/sin^2a+cos^4b/cos^2a
=A²/B+(1-B)²/(1-A)
=A²/A+(1-A)²/(1-A)
=A+1-A
=1
∴ 等式成立.

证明:令:X=cos²a, Y=cos²b
由题得:(1-X)²/(1-Y)+X²/Y=1
解之得: X=Y
所以,sin^4b/sin^2a+cos^4b/cos^2a=(1-Y)²/(1-X)+Y²/X
由上证...

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证明:令:X=cos²a, Y=cos²b
由题得:(1-X)²/(1-Y)+X²/Y=1
解之得: X=Y
所以,sin^4b/sin^2a+cos^4b/cos^2a=(1-Y)²/(1-X)+Y²/X
由上证得:Y=X,故将上式中的"Y“换成"X"
得:sin^4b/sin^2a+cos^4b/cos^2a=(1-X)²/(1-X)+X²/X=1-X+X=1

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