Question

Tan^2A=1+2tan^B prove that 2cos^2A=cos^2B

Answer 1

tan^2 A = 1 + 2tan^2 B 
=> sin^2 A / cos^2 A = 1 + 2 sin^2 B / cos^2 B 
=> sin^2 A / cos^2 A = (cos^2 B + 2sin^2 B) / cos^2 B 
=> sin^2 A cos^2 B = cos^2 A (cos^2 B + 2 sin^2 B) 
=> sin^2 A (1 - sin^2 B) = (1 - sin^2 A) (1 - sin^2 B + 2 sin^2 B) 
=> sin^2 A - sin^2 A sin^2 B = (1 - sin^2 A) (1 + sin^2 B) 
=> sin^2 A - sin^2 A sin^2 B = 1 - sin^2 A + sin^2 B - sin^2 A sin^2 B 
=> 2 sin^2 A = 1 + sin^2 B 

 

2 - 2 sin^2 A = 2 -(1 + sin^2 B)

2 cos^2 A = cos^2 B