Sum, Difference, and Multiple-Angle Formulas · Challenge problem

Angles $\alpha$ and $\beta$ lie strictly between $0$ and $\dfrac{\pi}{2}$, and $\cos\alpha\cos\beta=\dfrac13$ while $\sin\alpha\sin\beta=\dfrac16$. Find $\tan^2(\alpha+\beta)+\tan^2(\alpha-\beta)$.