（1）、2sin(A/2)-cos(B+C)
=2sin(A/2)+cosA
=2sin(A/2)+1-2(sinA/2)^2,
设sinA/2=t,
f(t)=-2t^2+2t+1
=-2(t^2-1+1/4-1/2-1/4)
=-2(t-1/2)^2+3/2,
当t=1/2时,有极大值3/2,
sin(A/2)=1/2,A/2=30度,
A=60度,
α=60°,
（2）、S△ABC=bcsinA/2
根据正弦定理,
b/sinB=a/sinA=c/sinC,
b=a*(sinB/sinA)=4sinB/√3,
c=a*(sinC/sinA)=4sinC/√3,
S△ABC=8sinBsinC/3
根据积化和差公式,
sinBsinC=-[cos(B+C)]/2+[cos(B-C)]/2,
由2sin(A/2)-cos(B+C）=3/2,
cos(B+C)=-1/2,
当B=C时,[cos(B-C)]/2有极大值1/2,
sinBsinC=-（-1/2）/2+1/2=3/4,
S△ABC=8sinBsinC/3=（8*3/4）/3=2.
△ABC的面积最大值为2.