What is the value of (cos 40° - cos 140°)/(sin 80° + sin 20°)?

Solve the numerator,

⇒ cos 40° - cos140° = - 2 sin[(140° + 40°)/2] × sin[(40° - 140°)/2]

⇒ cos 40° - cos140° = 2 sin[(140° + 40°)/2] × sin[(140° - 40°)/2]

⇒ cos 40° - cos 140° = 2sin90°.sin50°

⇒ cos 40° - cos 140° = 2sin50°

Solve the denominator,

sin 80° + sin 20° = 2 sin[(80° + 20°)/2] × cos[(80° - 20°)/2]

⇒ sin 80° + sin 20° = 2sin50°.cos30°

⇒ sin80° + sin20° = 2sin50° × (√3/2)

Replacing the respective value in the given equation:

(cos40° - cos140°)/(sin80° + sin20°) = (2sin50°)/(2sin50° × √3/2) = 2/√3

B is correct choice.