已知tan^2A=2tan^B+1求证sin^2B=2sin^2A-1

tan^2A=2tan^B+1
(sinA/cosA)^2=2(sinB/cosB)^2+1
sin^2A/cos^2A=2sin^2B/cos^2B+1
sin^2A*cos^2B=2sin^2Bcos^2A+cos^2Acos^2B
sin^2A(1-sin^2B)=2sin^2B(1-sin^2A)+(1-sin^2A)(1-sin^2B)
sin^2A-sin^2Asin^2B=2sin^2B-2sin^2Asin^2B+1-sin^2A-sin^2B+sin^2Asn^2B
sin^2B=2sin^2A-1