Proof of Multiple angle trigonometri ratios If $\tan^2{A}=1+2\tan^2B$ then prove that $\cos2B=1+2\cos2A$
I could not relate from $\tan$ to $\cos$

$$\frac{\sin^2 A}{\cos^2A}=1+2\frac{\sin^2 B}{\cos^2B}$$
using the identity $$\tan A = \frac{\sin A}{\cos A}$$
now using $$\cos^2A+\sin^2A = 1$$
$$\frac{1-\cos^2 A}{\cos^2 A} = 1+ 2\frac{1-\cos^2 B}{\cos^2 B}$$
$$\frac{1}{\cos^2 A}-1 = 1+ \frac{2}{\cos^2 B} -2$$
$$2\cos^2 A=\cos^2 B$$
and using the identity $$\cos^2 A = \frac{1+ \cos 2A}{2}$$
$$1+\cos 2A = \frac{1+\cos 2B}{2}$$
and
$$1+2\cos 2A = \cos 2B$$
