cos(2B)+3cos(A+C)+2=cos(2B)-3cosB+2=2cosB^2-3cosB+1
=(cosB-1)(2cosB-1)=0,即：cosB=1/2或cosB=1(不和题意,舍去)
故：B=π/3
m=(sqrt(3)a,-c),n=(cosA,sinC),故：m·n=sqrt(3)a*cosA-c*sinC
而：a/sinA=c/sinC=b/sinB=2=2R,R是外接圆半径
即：a=2sinA,c=2sinC
故：m·n=sqrt(3)a*cosA-c*sinC=2sqrt(3)sinA*cosA-2sinC*sinC
=sqrt(3)sin(2A)-(1-cos(2C))=sqrt(3)sin(2A)+cos(2C)-1
=sqrt(3)sin(2A)+cos(4π/3-2A)-1
=sqrt(3)sin(2A)-cos(2A)/2-sqrt(3)sin(2A)/2-1
=sqrt(3)sin(2A)/2-cos(2A)/2-1
=sin(2A-π/6)-1,0