Given, $\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ \cos (\alpha+\beta) & -\sin (\alpha+\beta) & 1\end{array}\right|$
$\left[\right.$ Applying $\left.\mathrm{R}_{3} \rightarrow \mathrm{R}_{3}-\mathrm{R}_{1}(\cos \beta)+\mathrm{R}_{2}(\sin \beta)\right]$ $=\left|\begin{array}{ccc}\cos \alpha & -\sin \alpha & 1 \\ \sin \alpha & \cos \alpha & 1 \\ 0 & 0 & 1+\sin \beta-\cos \beta\end{array}\right|$ $=(1+\sin \beta-\cos \beta)\left(\cos ^{2} \alpha+\sin ^{2} \alpha\right)$ $=1+\sin \beta-\cos \beta$, which is independent of $\alpha .$