Question

if tan^2A=1+2tan^2B then prove that cos^2B=2cos^2A

Answer 1

Given: The trigonometric term tan^2 A = 1 + 2tan^2 B

To find: Prove that cos^2 B = 2cos^2 A

Solution:

• Now we have given: tan^2 A = 1 + 2tan^2 B  

• Now converting tan in terms of sin and cos, we get:

          sin^2 A / cos^2 A = 1 + 2 sin^2 B / cos^2 B  

• Now solving further, we get:

          sin^2 A / cos^2 A = (cos^2 B + 2sin^2 B) / cos^2 B  

• Now cross multiplying, we get:

          sin^2 A cos^2 B = cos^2 A (cos^2 B + 2 sin^2 B)  

• Now converting all the terms to sin, we get:

          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  

 

Answer:

       Hence we have proved in solution that 2 sin^2 A = 1 + sin^2 B .