2.15. The dihedral angle at the base of a regular $n$-sided pyramid is $\alpha$. Find the dihedral angle between adjacent lateral faces.
2.15. Let $S$ be the area of a lateral face, $h$ the height of the pyramid, $a$ the side of the base, $\varphi$ the required angle. The area of the projection on the bisector plane of the dihedral angle between adjacent lateral faces for each of these faces is $S \cos (\varphi / 2)$; on the other hand, it is equal to $a h \sin (\pi / n) / 2$. It is also clear that the area of the projection of a lateral face onto a plane passing through its base perpendicular to the base of the pyramid is $S \sin \alpha$; on the other hand, it is equal to $ah / 2$. Therefore, $\cos (\varphi / 2)=\sin \alpha \sin (\pi / n)$.