The value of the determinant $\left|\begin{array}{rrr}a \cos \theta & b \sin \theta & 0 \\ -b \sin \theta & a \cos \theta & 0 \\ 0 & 0 & c\end{array}\right|$ is:

$\left(a^2 \cos ^2 \theta+b^2 \sin ^2 \theta\right) c$

The correct answer is Option (2) - $\left(a^2 \cos ^2 \theta+b^2 \sin ^2 \theta\right) c$

expanding determinant along $R_3$ we get

$c\left(a^2 \cos ^2 \theta+b^2 \sin ^2 \theta\right)$