12.291. The cosine of the angle between the lateral edges of a regular quadrilateral pyramid, not lying in the same face, is $k$. Find the cosine of the dihedral angle at the vertex of the pyramid.
## Solution.

Let $S O$ be the height of the regular pyramid $S A B C D$ (Fig. 12.153), $\cos \angle A S C=k$.

If $\angle A S C=\alpha$, then $\angle D S O=\angle C S O=\frac{\alpha}{2}$.

Consider the plane $S O C$. $S D$ is inclined to this plane, $S O$ is the projection of $S D$ onto this plane, and $S C$ is a line lying in the plane $S O C$.

Then $\cos \angle D S C=\cos \angle D S O \cdot \cos \angle C S O=\cos ^{2} \frac{\alpha}{2}=\frac{1+\cos \alpha}{2}=\frac{1+k}{2}$.

Answer: $\frac{1+k}{2}$.