If $$ \tan A = n \tan B  and   \sin A = m \sin B,$$  then the value of $$ \cos^2 A$$ is

Given that $$ \tan A = n \tan B $$ and  $$\sin A = m \sin B$$ ---- (1)

=> $$\frac{\sin A}{\cos A}$$ = n$$\frac{\sin B}{\cos B}$$

=> $$\frac{m\sin B}{\cos A}$$ = n $$\frac{\sin B}{\cos B}$$

=> $$\frac{\cos A}{\cos B}$$ = $$\frac {m}{n}$$ ----- (2)

Squaring equation (1), we get

=> $$\sin^2 A = m^2 \sin^2 B$$

=> $$ 1-\cos^2 A = m^2 (1-cos ^2 B) $$

=> $$ cos ^2 B = \frac {m^2 -1 + \cos^2 A}{m^2}$$ ---- (3)

Squaring equation (2) and substituting equation (3) in equation (2), we get

=> $$ cos^2 A = [\frac{m^2}{n^2}][\frac {m^2 -1 + \cos^2 A}{m^2}]$$

=> $$ n^2 cos^2 A= m^2 -1 + \cos^2 A $$

=> $$ cos ^2 A = \frac {m^2 -1}{n^2 -1} $$