Ctg 80*ctg 70 + ctg 70*ctg 30 + ctg 30*ctg 80 =
= ctg (90-10)*ctg (90-20) + ctg (90-20)*√3 + √3*ctg(90-10) =
= tg 10*tg 20 + √3*tg 20 + √3*tg 10 = (sin 10*sin 20) / (cos 10*cos 20) + √3*(tg 10 + tg 20) =
= (sin 10*sin 20) / (cos 10*cos 20) + √3*sin(10 + 20) / (cos 10*cos 20) =
= (sin 10*sin 20 + √3*sin 30) / (cos 10*cos 20) =
= [1/2*(cos 10 - cos 30) + √3/2] / [1/2*(cos 10 + cos 30] =
= (cos 10 - √3/2 + √3) / (cos 10 + √3/2) = (cos 10 + √3/2) / (cos 10 + √3/2) = 1