First, let's convert all angles to radians.

80° = 80(π/180) = 4π/9
40° = 40(π/180) = 2π/9

Now we can calculate the value of cos80° and sin40°

cos(80°) = cos(4π/9) ≈ 0.1736
sin(40°) = sin(2π/9) ≈ 0.3420

Now substitute these values back into the expression:

cos80°÷(cos40°+sin40°) = 0.1736 / (0.1736 + 0.3420)
= 0.1736 / 0.5156
≈ 0.3367

Therefore, the value of cos80°÷(cos40°+sin40°) is approximately 0.3367.