Question

Помогите решить:

 

$cos10cdot cos30cdot cos50cdot cos70 = ?$

Answer 1

$cos10cdot cos30cdot cos50cdot cos70 = \ =frac{sqrt{3}}{2}cdotfrac{1}{2}(cos(50-10)+cos(50+10))cdot cos(90-20)= \ = frac{sqrt{3}}{4}(cos40+cos60)cdot sin20= \ =frac{sqrt{3}}{4}(cos40cdot sin20+frac{1}{2}cdot sin20)= \ =frac{sqrt{3}}{4}(frac{1}{2}(sin(20-40)+sin(20+40)+frac{1}{2}sin20)= \ =frac{sqrt{3}}{4}(frac{1}{2}(-sin20+sin60)+frac{1}{2}sin20)= \ =frac{sqrt{3}}{4}(-frac{1}{2}sin20+frac{sqrt{3}}{4}+frac{1}{2}sin20)=frac{3}{16}$