Answer to Question #88933 in Trigonometry for Manu Chemjong

Question

If tan^2a=1+2tan^2b,then prove that cos 2b=1+2cos2a

Answer 1

The identity: tan²a = $\frac{1}{cos^{2}a}$ - 1; tan²b = $\frac{1}{cos^{2}b}$ - 1;

$\implies$ $\frac{1}{cos^{2}a}$ - 1 = 1 + 2($\frac{1}{cos^{2}b}$ - 1);

$\implies$ $\frac{1}{cos^{2}a}$ = $\frac{2}{cos^{2}b}$ ; cos²b = 2cos²a;

The identity: cos²b = $\frac{1+cos2b}{2}$; cos²a = $\frac{1+cos2a}{2}$ ;

$\implies$ $\frac{1+cos2b}{2}$ = 1 + cos2a;

$\implies$ cos2b = 1 + 2cos2a.

Answer: from the relation tan²a = 1 + 2tan²b, follows: cos2b = 1 + 2cos2a.