[tex]E = \dfrac{\sin 40^{\circ}+\sin 20^{\circ}}{\cos 40^{\circ}+\cos 20^{\circ}} = \dfrac{2\sin\frac{40^{\circ}+20^{\circ}}{2}\cos \frac{40^{\circ}-20^{\circ}}{2}}{2\cos\frac{40^{\circ}+20^{\circ}}{2}\cos \frac{40^{\circ}-20^{\circ}}{2}} =[/tex]
[tex]= \dfrac{2\sin\frac{60^{\circ}}{2}\cos \frac{20^{\circ}}{2}}{2\cos\frac{60^{\circ}}{2}\cos \frac{20^{\circ}}{2}} = \dfrac{\sin 30^{\circ}\cos 10^{\circ}}{\cos 30^{\circ}\cos 10^{\circ}} = \dfrac{\sin 30^{\circ}}{\cos 30^{\circ}}=[/tex]
[tex]= \text{tg}\, 30^{\circ} = \boxed{\dfrac{\sqrt{3}}{3}}[/tex]
