<s>[INST] 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 $\cos$
 [/INST] Add $1$ to both sides of the tangent condition.
We get $\tan^2A+1=2+2\tan^2B$.
Simplifying with the identity $\tan^2\theta+1=\sec^2\theta$, we have:
$$\sec^2A=2\sec^2B$$
Taking the reciprocals of both sides and multiplying by $4$, we get:
$$4\cos^2A=2\cos^2B$$
Remembering the identity $2\cos^2\theta-1=\cos2\theta$, we can subtract $1$ from each side to get:
$$2(2\cos^2A-1)+1=2\cos^2B-1$$
which simplifies to:
$$2\cos2A+1=\cos2B$$
</s>