Israel winter camp 2013

[arqady]

Prove that:

$$\tan31^{\circ}\tan35^{\circ}\tan41^{\circ}\tan67^{\circ}\tan77^{\circ}=\tan75^{\circ}$$

[Galua]

a=tan 31 tan 41=$\frac{cos 10-cos72}{cos10+cos72}$ b=tan 67tan 77=$\frac{cos 10+cos 36}{cos10-cos 36}$ ab=$\frac{cos^2 10+0.5cos 10 -0.25}{cos^2-0.5cos 10-0.25}$=$\frac{1+2cos 10+cos 20}{1-2cos 20+2cos 10}$= $\frac{1+4cos 15cos 5}{1-4sin 15sin 5}$ abtan 15=$\frac{sin 15+cos 5}{cos15-sin 5}$=cot 35
Отсюда следует требуемое.