【Example 4. 22】(1) Find $\sin ^{2} 20^{\circ}+\cos ^{2} 80^{\circ}+\sqrt{3} \sin 20^{\circ} \cos 80^{\circ}$. (1999 College Entrance Examination Question)
(2) Find $\sin ^{2} 20^{\circ}+\cos ^{2} 50^{\circ}+\sin 20^{\circ} \cos 50^{\circ}$ (1995 College Entrance Examination Question)
(1) Original expression $=\sin ^{2} 20^{\circ}+\sin ^{2} 10^{\circ}-2 \sin 20^{\circ} \sin 10^{\circ} \cos 150^{\circ}=$ $\sin ^{2} 150^{\circ}=\frac{1}{4}$.
(2) Original expression $=\sin ^{2} 20^{\circ}+\sin ^{2} 40^{\circ}-2 \sin 20^{\circ} \sin 40^{\circ} \cos 120^{\circ}=$ $\sin ^{2} 120^{\circ}=\frac{3}{4}$.

Note the function with the same name, $20+10+150=180,20+40+120=180$, which can be used in a triangle with $2 R=1$ using the Law of Sines and the Law of Cosines.