9. Let $A=\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}$. Determine the value of $100 A$.
9. Answer: 75
Solution. Let
$$
B=\sin ^{2} 10^{\circ}+\sin ^{2} 50^{\circ}-\cos 40^{\circ} \cos 80^{\circ} .
$$

Then
$$
A+B=2-\cos 40^{\circ}
$$
and
$$
\begin{aligned}
A-B=\cos 20^{\circ}+\cos 100^{\circ} & +\cos 120^{\circ}=2 \cos 60^{\circ} \cos 40^{\circ}+\cos 120^{\circ} \\
& =\cos 40^{\circ}-\frac{1}{2}
\end{aligned}
$$

Thus $2 A=\frac{3}{2}$ and $100 A=75$.