$3.427 \cos 50^{\circ}+8 \cos 200^{\circ} \cdot \cos 220^{\circ} \cdot \cos 80^{\circ}=2 \sin ^{2} 65^{\circ}$.
Solution. $\cos 50^{\circ}+8 \cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 80^{\circ}=1-\cos 130^{\circ} \Leftrightarrow$

$\Leftrightarrow 8 \sin 20^{\circ} \cdot \cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 80^{\circ}=\sin 20^{\circ} \Leftrightarrow 4 \sin 40^{\circ} \cdot \cos 40^{\circ} \cdot \cos 80^{\circ}=\sin 20^{\circ} \Leftrightarrow$ $\Leftrightarrow 2 \sin 80^{\circ} \cdot \cos 80^{\circ}=\sin 20^{\circ} \Leftrightarrow \sin 160^{\circ}=\sin 20^{\circ}$.