If cos α + cos β = 0 = sin α + sin β, α ≠ β then what is a value of cos 2α + cos 2β + 2 cos(α + β)?

cos 2α + cos 2β = 2 cos(α+β) cos(α-β). Adding 2cos(α+β): 2cos(α+β)[cos(α-β) + 1]. From given: cos α = -cos β and sin α = -sin β, so α = π + β (mod 2π), making cos(α+β) = cos(π+2β) = -cos 2β. Also cos(α-β) = cos π = -1. So 2cos(α+β)[-1+1] = 0.