已知sin^2A+cos^2A*sin^2B*cos^2C=sin^2B,求证tan^2A=sin^2C*tan^2B,请大神速来回答啊sin^2A的意思是(sin A)的平方,中间的*是×(乘)

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已知sin^2A+cos^2A*sin^2B*cos^2C=sin^2B,求证tan^2A=sin^2C*tan^2B,请大神速来回答啊sin^2A的意思是(sin A)的平方,中间的*是×(乘)

已知sin^2A+cos^2A*sin^2B*cos^2C=sin^2B,求证tan^2A=sin^2C*tan^2B,请大神速来回答啊sin^2A的意思是(sin A)的平方,中间的*是×(乘)
已知sin^2A+cos^2A*sin^2B*cos^2C=sin^2B,求证tan^2A=sin^2C*tan^2B,请大神速来回答啊
sin^2A的意思是(sin A)的平方,中间的*是×(乘)

已知sin^2A+cos^2A*sin^2B*cos^2C=sin^2B,求证tan^2A=sin^2C*tan^2B,请大神速来回答啊sin^2A的意思是(sin A)的平方,中间的*是×(乘)
sin²A+cos²Asin²Bcos²C=sin²B
sin²A=sin²B(1-cos²Acos²C)
sin²A=sin²B[sin²A+cos²A-cos²Acos²C]
sin²A=sin²B[sin²A+cos²A(1-cos²C)]
sin²A=sin²B[sin²A+cos²Asin²C]
sin²A-sin²Bsin²A=sin²Bcos²Asin²C
sin²A(1-sn²B)=sin²Bcos²Asin²C
sin²Acos²B=sin²Bcos²Asin²C
sin²A/cos²A=(sin²B/cos²B)sin²C)
tan²A=tan²Bsin²C

因为sin^2A+cos^2A*sin^2B*cos^2C=sin^2B
cos^2A*sin^2B*cos^2C=sin^2B-sin^2A
cos^2C=(sin^2B-sin^2A)/(cos^2A*sin^2B)
cos^2C=(1-sin^2A/sin^2B)/cos^2A
所以sin^2C=1-(1-sin^2A/sin^2B)/cos^2A
...

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因为sin^2A+cos^2A*sin^2B*cos^2C=sin^2B
cos^2A*sin^2B*cos^2C=sin^2B-sin^2A
cos^2C=(sin^2B-sin^2A)/(cos^2A*sin^2B)
cos^2C=(1-sin^2A/sin^2B)/cos^2A
所以sin^2C=1-(1-sin^2A/sin^2B)/cos^2A
=1+sin^2A/(sin^2B*cos^2A)-1/cos^2A
=(sin^2B*cos^2A+sin^2A-sin^2B)/(sin^2B*cos^2A)
=[(1-cos^2B)*(1-sin^2A)+sin^2A-1+cos^2B]/(sin^2B*cos^2A)
=(sin^2A*cos^2B)/(sin^2B*cos^2A)
=tan^2A/tan^2B
tan^2A=sin^2C*tan^2B

收起

条件不够啊

被楼上那位虐爆掉。。。

慢慢算就行了。
sin^2A+cos^2A*sin^2B*cos^2C=sin^2B可以将cos^2C=1-sin^2C,然后化简成sin^2C=(sin^2+cos^2A*sin^2B-sin^2B)/(cos^2*sin^2B);然后将分子中的cos^2A=1-sin^2A.然后就可以得出
sin^2C=(sin^2A-sin^2A*sin^2B)/(cos^2*sin^2B);然后将分子提出sin^2A,所以Sin^2C=tan^2A/tan^2B,所以得证