A Pyramid Inscribed in a Cylinder

As shown in the figure, $ABCD$ is a quadrilateral inscribed in the base circle of a cylinder, $AC$ is a diameter of that base circle, $PC$ is a generatrix of the cylinder (so $P$ lies on the top rim, directly above $C$), and $E$ is the point of intersection of $AC$ and $BD$. Moreover $AB = AD$ and $\angle BAD = 60^{\circ}$.

1. Let $V_1$ be the volume of the cylinder and $V_2$ the volume of the pyramid $P\text{-}ABCD$. Find $\dfrac{V_1}{V_2}$. 2. Suppose the point $F$ lies on the segment $AP$ with $PA = 4PF$, and $PC = 4CE$. Find the cosine of the dihedral angle $F\text{-}CD\text{-}P$.