f（x）=2cosx/2(根号3sinx/2+cosx/2)-1

（1）f(x)=2√3cosx/2sinx/2+2cos²x/2-1=√3sinx+cosx=2sin(x+π/6),最小正周期T=2π。（2）由f(α)=2得2sin(α+π/6)=2,∴sin(α+π/6)=1,α+π/6=π/2,∴α=π/3,又f(β)=8/5,即sin(β+π/6)=4/5,cos(β+π/6)=3/5,f(α＋β)=2sin(α+β＋π/6)=2sin(π/3+β＋π/6)=2sinπ/3cos(β+π/6)+2cosπ/3sin(β＋π/6)=√3×3/5＋2×1/2×4/5=3√3/5+4/5=(3√3+4)/5