Consider a triangle ABC,
b= a*cosC + c*cosA
c= a*cosB + b*cosA

Adding both the equations and upon simplification,
 we get
(b+c)*(1-cosA)=a*(cosB+cosC)
but in a triangle, b+c>a 
=> (b+c)*(1-cosA)>a*(1-cosA)
=> a*(cosB+cosC)>a*(1-cosA)
=> cosA+cosB+cosC>1.

Hence Proved