热心网友
cosx+cos2x+cos3x+......+cosnx=1/(2sin(x/2))*[2cosxsin(x/2)+2cos2xsin(x/2)+......+2cosnxsin(x/2)]括号中的数列的和等于“[sin(3x/2)-sin(x/2)]+[sin(5x/2)-sin(3x/2)]+[sin(7x/2)-sin(5x/2)]+......+{sin[(2n+1)x/2]-sin[(2n-1)x/2]}=sin[(2n+1)x/2]-sin(x/2)=2cos(n+1)xsin(nx/2)所以，原式=cos(n+1)xsinnx/sin(x/2).