证明tan^2x=2*tan^2y+1,求证sin^2y=2sin^2x-1
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证明tan^2x=2*tan^2y+1,求证sin^2y=2sin^2x-1
![证明tan^2x=2*tan^2y+1,求证sin^2y=2sin^2x-1](/uploads/image/z/17574802-34-2.jpg?t=%E8%AF%81%E6%98%8Etan%5E2x%3D2%2Atan%5E2y%2B1%2C%E6%B1%82%E8%AF%81sin%5E2y%3D2sin%5E2x-1)
tan^2x=2*tan^2y+1;
sin^2x=tan^2x/(1+tan^2x);
sin^2y=tan^2y/(1+tan^2y);
sin^2y-2sin^2x+1=tan^2y/(1+tan^2y)-2*tan^2x/(1+tan^2x)+1
=tan^2y/(1+tan^2y)-2*(2*tan^2y+1)/(1+2*tan^2y+1)+1
=(tan^2y-2tan^2y-1)/(1+tan^2y)+1=0;
所以sin^2y=2sin^2x-1.
sin^2x=tan^2x/(1+tan^2x);
sin^2y=tan^2y/(1+tan^2y);
sin^2y-2sin^2x+1=tan^2y/(1+tan^2y)-2*tan^2x/(1+tan^2x)+1
=tan^2y/(1+tan^2y)-2*(2*tan^2y+1)/(1+2*tan^2y+1)+1
=(tan^2y-2tan^2y-1)/(1+tan^2y)+1=0;
所以sin^2y=2sin^2x-1.
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