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

$a^2$

$\left|\begin{array}{rr}a \cos \theta & a \sin \theta \\ -a \sin \theta & a \cos \theta\end{array}\right|$

$=(a \cos \theta)(a \cos \theta)-(-a \sin \theta)(a \sin \theta)$

$=a^2 \cos ^2 \theta+a^2 \sin ^2 \theta$

$\Delta=a^2\left(\sin ^2 \theta+\cos ^2 \theta\right)$

as $\sin ^2 \theta+\cos ^2 \theta = 1$

$=a^2$