If $f\left(\theta \right) = \left|\begin{array}{ccc}{\cos }^2\theta & \cos \theta \sin \theta & -\sin \theta \\ \cos \theta \sin \theta & {\sin }^2\theta & \cos \theta \\ \sin \theta & -\cos \theta & 0\end{array}\right|$ then $\mathrm{f}\left(\frac{\mathrm{\pi }}{24}\right) =$

$\begin{array}{rcl}\mathrm{f}\left(\mathrm{\theta }\right)& =& {\cos }^2\mathrm{\theta }.{\cos }^2\mathrm{\theta }+{\cos }^2\mathrm{\theta } {\sin }^2\mathrm{\theta }-\mathrm{sin\theta }(-{\cos }^2\mathrm{\theta } \mathrm{sin\theta }-{\sin }^3\mathrm{\theta })\\ & =& {\cos }^4\mathrm{\theta }+{\cos }^2\mathrm{\theta }-{\sin }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }.{\cos }^2\mathrm{\theta }+{\sin }^4\mathrm{\theta }\\ & =& {\cos }^2\mathrm{\theta }({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta })+{\sin }^2\mathrm{\theta }({\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta })\\ & =& {\cos }^2\mathrm{\theta }+{\sin }^2\mathrm{\theta }\\ & =& 1\\ \therefore \mathrm{f}\left(\frac{\mathrm{\pi }}{24}\right)& =& 1\end{array}$