Evaluate the following determinants: $(i)$ $\left|\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right|$ (ii) $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right|$

(N/A) $(i)$ $\left|\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right| = (\cos \theta)(\cos \theta) - (-\sin \theta)(\sin \theta) = \cos^{2} \theta + \sin^{2} \theta = 1$
(ii) $\left|\begin{array}{cc}x^{2}-x+1 & x-1 \\ x+1 & x+1\end{array}\right| = (x^{2}-x+1)(x+1) - (x-1)(x+1)$
$= (x^{3} + x^{2} - x^{2} - x + x + 1) - (x^{2} - 1)$
$= (x^{3} + 1) - x^{2} + 1$
$= x^{3} - x^{2} + 2$