Show that the determinant $\displaystyle \left|\begin{array}{ccc}x & \sin \theta & \cos \theta \\ -\sin \theta & -x & 1 \\ \cos \theta & 1 & x\end{array}\right|$ is independent of $\displaystyle \theta$.

$\displaystyle \Delta = \begin{vmatrix} x & \sin\theta & \cos\theta \\ -\sin\theta & -x & 1 \\ \cos\theta & 1 & x \end{vmatrix}$\[= x(-x^{2}-1) - \sin\theta\,(-x\sin\theta - \cos\theta) + \cos\theta\,(-\sin\theta + x\cos\theta)\]\[= -x^{3} - x + x\sin^{2}\theta + \sin\theta\cos\theta - \cos\theta\sin\theta + x\cos^{2}\theta\]\[= -x^{3} - x + x\]\[= -x^{3}, \text{ independent of } \theta\]