(b+c)cosA+(c+a)cosB+(a+b)cosC
=(b+c)*(b^2+c^2-a^2)/2bc+(a+c)*(c^2+a^2-b^2)/2ac+(a+b)*(a^2+b^2-c^2)/2ab
=(b^3+b^2*c+bc^2+c^3-a^2*b-a^2*c)/2bc+(ac^2+c^3+a^3+a^2*c-ab^2-b^2*c)/2ac+(a^3+a^2*b+ab^2+b^3-ac^2-bc^2)/2ab
=(ab^3+ab^2*c+abc^2+ac^3-a^3*b-a^3*c+abc^2+bc^3+a^3*b+a^2*bc-ab^3-b^3*c+a^3*c+a^2bc+ab^2*c+b^3*c-ac^3-bc^3)/2abc
=abc(b+c+c+a+a+b)/2abc
=2abc(a+b+c)/2abc
=a+b+c