(sin^2 80°)(1+√3tan10°)/4(cos40°-cos^3 40°)
=(sin^2 80°)(1+√3tan10°)/[4cos40°(1-cos^2 40°) ]
=(sin^2 80°)(1+√3tan10°)/(4cos40°*sin^2 40°)
= (sin^80°)(1+√3tan10°)/2sin40°
= (cos10°)(1/2+√3/2tan10°)/sin40°=(1/2cos10°+√3/2sin10°)/sin40°
=(sin30°cos10°+cos30°sin10°)/sin40°=sin(30°+10°)/sin40°=1