(b) $\Delta = \,\left| {\,\begin{array}{*{20}{c}}1&{\cos (\beta – \alpha )\,}&{\cos (\gamma – \alpha )}\\{\cos (\alpha – \beta )\,}&1&{\cos (\gamma – \beta )}\\{\cos (\alpha – \gamma )}&{\cos (\beta – \gamma )\,}&1\end{array}\,} \right|$

$=\left|\begin{array}{ccc}\cos ^2 \alpha+\sin ^2 \alpha & \cos \beta \cos \alpha+\sin \beta \sin \alpha & \cos \alpha \cos \gamma+\sin \alpha \sin \gamma \\ \cos \alpha \cos \beta+\sin \alpha \sin \beta & \cos ^2 \beta+\sin ^2 \beta & \cos \beta \cos \gamma+\sin \beta \sin \gamma \\ \cos \alpha \cos \gamma+\sin \alpha \sin \gamma & \cos \beta \cos \gamma+\sin \beta \sin \gamma & \cos ^2 \beta+\sin ^2 \beta\end{array}\right|$

$=\left|\begin{array}{ccc}\cos \alpha & \sin \alpha & 0 \\ \cos \beta & \sin \beta & 0 \\ \cos \gamma & \sin \gamma & 0\end{array}\right| \cdot\left|\begin{array}{lll}\cos \alpha & \sin \alpha & 0 \\ \cos \beta & \sin \beta & 0 \\ \cos \gamma & \sin \gamma & 0\end{array}\right|=\left|\begin{array}{lll}\sin \alpha & \cos \alpha & 0 \\ \sin \beta & \cos \beta & 0 \\ \sin \gamma & \cos \gamma & 0\end{array}\right|^2$