因为：a/sinA=b/sinB=c/sinC=2R
所以：a^2=4R^2*sinA b^2=4R^2*sinB c^2=4R^2*sinC
所以:
(a^2 - b^2)/(cosA+cosB)
=4R^2*(sin^2A-sin^2B)/[2cos(A+B)/2*cos(A-B)/2]
=4R^2*(sinA+sinB)(sinA-sinB)/[2cos(A+B)/2*cos(A-B)/2]
=4R^2*2sin[(A+B)/2]cos[(A-B)/2]*2sin[(A-B)/2]*cos[(A+B)/2]/[2cos(A+B)/2*cos(A-B)/2]
=4R^2*sin(A/2+B/2)*sin(A/2-B/2)
=4R^2*[sin^2(A/2)-sin^2(B/2)]
所以：
(a^2 - b^2)/(cosA+cosB)+(b^2 - c^2)/(cosB+cosC)+(c^2 - a^2)/(cosC+cosA)
=4R^2*[sin^2(A/2)-sin^2(B/2)+sin^2(B/2)-sin^2(C/2)+sin^2(C/2)-sin^2(A/2)]
=0