A little Problem in Trigonometry (Multiple Angle) If $\tan^2 \theta = 1 + 2\tan^2 \phi$, show that $\cos 2\phi = 1 + 2\cos2\theta$.
What I have done..
$$\implies \tan^2 \theta = 1 + 2\tan^2 \phi\\
\implies 1 + \tan^2 \theta = 2 + 2\tan^2 \phi\\
\implies 1 + \tan^2 \theta = 2(1 + \tan^2 \phi)\\
\implies \sec^2 \theta = 2(\sec^2 \phi)$$

Your steps are fine you just need to do the following (I'm putting your steps first):
\begin{align*}
\tan^{2}\theta & = 1+2 \tan^{2}\phi\\
1+\tan^{2}\theta & = 2+2 \tan^{2}\phi\\
\sec^2 \theta & = 2 \sec^2 \phi\\
\text{Do the following to continue:}\\
\frac{1}{\cos^2 \theta} & =  \frac{2}{\cos^2 \phi}\\
\cos^2 \phi & = 2 \cos^2 \theta\\
\text{using the half angle formula}\\
\frac{1+\cos 2 \phi}{2} & = 1+\cos2 \theta\\
1+\cos 2 \phi & = 2(1+\cos2 \theta)\\
\cos 2\phi & = 1 +2\cos 2\theta.
\end{align*}
