The determinant $\Delta=\left|\begin{array}{ccc}\cos (\theta+\varphi) & -\sin (\theta+\varphi) & \cos 2 \varphi \\ \sin \theta & \cos \theta & \sin \varphi \\ -\cos \theta & \sin \theta & \cos \varphi\end{array}\right|$ is

independent of $\theta$

Apply $R_1 \rightarrow R_1+R_2 \sin \phi-R_3 \cos \phi$

Then taking $2 \cos \phi$ common from $R_1$  & then apply $R_1 \rightarrow R_1+R_3$

Then $\Delta=2 \cos \phi$

Hence (2) is the correct answer.