Let $\Delta=\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \cos (\alpha+\delta) \\ \sin \beta & \cos \beta & \cos (\beta+\delta) \\ \sin \gamma & \cos \gamma & \cos (\gamma+\delta)\end{array}\right|$
Using $\cos (A+B)=\cos A \cos B-\sin A \sin B$
$\Delta=\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & \cos \alpha \cos \delta-\sin \alpha \sin \delta \\ \sin \beta & \cos \beta & \cos \beta \cos \delta-\sin \beta \sin \delta \\ \sin \gamma & \cos \gamma & \cos \gamma \cos \delta-\sin \gamma \sin \delta\end{array}\right|$
Applying $\mathrm{C}_3 \rightarrow \mathrm{C}_3+\sin \delta \times \mathrm{C}_1-\cos \delta \cdot \mathrm{C}_2$
$\Delta=\left|\begin{array}{ccc}\sin \alpha & \cos \alpha & 0 \\ \sin \beta & \cos \beta & 0 \\ \sin \gamma & \cos \gamma & 0\end{array}\right|=0$
By matrix method.