By sine law we know
#a/sinA=b/sinB=c/sinC=2R#
Now
1st part
#(b^2-c^2)cotA#
#=(4R^2sin^2B-4R^2sin^2C)cotA#
#=4R^2(1/2(1-cos2B)-1/2(1-cos2C)cotA#
#=4R^2xx1/2(cos2C-cos2B)cotA#
#=2R^2xx2sin(B+C)sin(B-C)cosA/sinA#
#=4R^2sin(pi-A)sin(B-C)cosA/sinA#
#=4R^2sinAsin(B-C)cosA/sinA#
#=4R^2sin(B-C)cosA#
#=4R^2(sinBcosCcosA-cosBsinCcosA)#
Similarly
2nd part #=(c^2-a^2)cotB#
#=4R^2(sinCcosAcosB-cosCsinAcosB)#
3rd part #=(a^2-b^2)cotC#
#=4R^2(sinAcosBcosC-cosAsinBcosC)#
Adding three parts we get
Whole expression
#(b^2-c^2)cotA+(c^2-a^2)cotB+(a^2-b^2)cotC=0#
