Question:
Show that $\left|\begin{array}{cc}\sin 10^{\circ} & -\cos 10^{\circ} \\ \sin 80^{\circ} & \cos 80^{\circ}\end{array}\right|=1$
Solution:
Let $\Delta=\left|\begin{array}{cc}\sin 10^{\circ} & -\cos 10^{\circ} \\ \sin 80^{\circ} & \cos 80^{\circ}\end{array}\right|$
$\Rightarrow \Delta=\sin 10^{\circ} \cos 80^{\circ}+\cos 10^{\circ} \sin 80^{\circ}$
$=\sin 10^{\circ} \cos \left(90^{\circ}-10^{\circ}\right)+\cos 10^{\circ} \sin \left(90^{\circ}-10^{\circ}\right) \quad[\because \cos \theta=\sin (90-\theta)]$
$\Rightarrow \Delta=\sin 10^{\circ} \sin 10^{\circ}+\cos 10^{\circ} \cos 10^{\circ}$
$=\sin ^{2} 10^{\circ}+\cos ^{2} 10^{\circ} \quad\left[\because \sin ^{2} \theta+\cos ^{2} \theta=1\right]$
$\Rightarrow \Delta=1$