∵f(x)是偶函数,∴f(-x)=f(x)
即√2cos(-x/2)-(a-1)sin(-x/2)=√2cos(x/2)-(a-1)sin(x/2)
即√2cos(x/2)+(a-1)sin(x/2)=√2cos(x/2)-(a-1)sin(x/2)
∴2(a-1)sin(x/2)=0,∴a=1
f(x)=√2cos(x/2),所以最小正周期为T=2π/(1/2)=4π