The value of the determinant

Solution:

Let $\Delta=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & \cos ^{2} 180^{\circ} \\ \cos 180^{\circ} & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$\Delta=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & \cos ^{2} 180^{\circ} \\ \cos 180^{\circ} & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ} & -1 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & (-1)^{2} \\ -1 & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ} & -1 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & 1 \\ -1 & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

Applying $R_{1} \rightarrow R_{1}+R_{2}$

$=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}-\sin ^{2} 33^{\circ} & -1+1 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & 1 \\ -1 & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}-\sin ^{2} 33^{\circ} & 0 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & 1 \\ -1 & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

Taking out $\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)$ common from $R_{1}$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & -1 & 0 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & 1 \\ -1 & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

Applying $C_{2} \rightarrow C_{2}+C_{1}$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & -1+1 & 0 \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ} & 1 \\ -1 & \sin ^{2} 33^{\circ}-1 & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & \cos ^{2} 33^{\circ}-1-\sin ^{2} 57^{\circ} & 1 \\ -1 & -\left(1-\sin ^{2} 33^{\circ}\right) & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & \cos ^{2} 33^{\circ}-1-\sin ^{2} 57^{\circ} & 1 \\ -1 & -\cos ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & \cos ^{2}\left(90^{\circ}-57^{\circ}\right)-1-\sin ^{2} 57^{\circ} & 1 \\ -1 & -\cos ^{2}\left(90^{\circ}-57^{\circ}\right) & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}-1-\sin ^{2} 57^{\circ} & 1 \\ -1 & -\sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & -1 & 1 \\ -1 & -\sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

Taking out $(-1)$ common from $C_{2}$

$=(-1)\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)\left|\begin{array}{ccc}1 & 0 & 0 \\ -\sin ^{2} 57^{\circ} & 1 & 1 \\ -1 & \sin ^{2} 57^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$

$=(-1)\left(\sin ^{2} 33^{\circ}-\sin ^{2} 57^{\circ}\right)(0) \quad\left(\because C_{2}\right.$ and $C_{3}$ are identical, $\therefore$ the value of determinant is zero)

$=0$

Hence, the value of the determinant $\Delta=\left|\begin{array}{ccc}\sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ} & \cos 180^{\circ} \\ -\sin ^{2} 57^{\circ} & -\sin ^{2} 33^{\circ} & \cos ^{2} 180^{\circ} \\ \cos 180^{\circ} & \sin ^{2} 33^{\circ} & \sin ^{2} 57^{\circ}\end{array}\right|$ is equal to $\underline{0}$.