sinα+sinβ=m,
两边平方,
(sinα)^2+2sinαsinβ+(sinβ)^2=m^2,.(1)
cosα+cosβ=√2,
两边平方,
（cosα）^2+2cosαcosβ+(cosβ)^2=2,.(2),
(1)+(2)式,
1+2cos(α-β）+1=m^2+2,
m^2=2cos(α-β）,
m<=√[2cos(α-β）],
2cos(α-β）最大值为2,
-√2<=m<=√2,
α-β=kπ（k∈Z）时,m有极值,（最大和最小）,
β=kπ+α
cosα+cos（kπ+α）=√2,
当k是偶数时,
cosα+cosα=√2,
cosα=√2/2,
α=2nπ±π/4,（n∈Z）,
k为奇数时,cosα-cosα=√2,无意义,
m取最小值时,sin(α+β）=sin( α+kπ+α)=sin(2α+kπ)=sin(π/2+kπ）=±1,