Solution: Let the square of cosα+cosβ =k have COS 2 (α)+COS 2 (β)+2 COS α * COS β = K2 (1).

Similarly, sin α+sin β = radical number 2, and the squares of both sides are sin2 (α)+sin2 (β)+2sinα * sinβ =1/2 (2).

(1)+(2) There are 2+2cosα * cosβ+2sinα * sinβ = k2+1/22+2cos (α-β) = k2+1/2 (3).

2cos (α-β) = k2+1/2-2 = k2-3/2cos (α-β) =1/2 * k2-3/4 because | cos (α-β) | "65438+.

So |1/2k2-3/4 | "1-kloc-0/"1/2k2-3/4 "1/4". =0

So 0 "1/2k 2" 7/40 "K 2" 7/2- radical number 14/2 "K "radical number 14/2.