CN-LEAGUE 2012 Problem 5

Suppose two regular pyramids with the same base $ABC$ : $P-ABC$ and $Q-ABC$ are circumscribed by the same sphere. If the angle formed by one of the lateral face and the base of pyramid $P-ABC$ is $\frac{\pi}{4}$ , find the tangent value of the angle formed by one of the lateral face and the base of the pyramid $Q-ABC$ .