A>0 B>0 C>0 A+B+C=π から sinA>0 sinB>0 sinC>0 なので
cosA+cosB+cosC-1
=2cos{(A+B)/2}cos{(A-B)/2}-2sin^2(C/2)
=2cos{(π-C)/2}cos{(A-B)/2}-2sin(C/2)sin{(π-A-B)/2}
=2sin(C/2)cos{(A-B)/2}-2sin(C/2)cos{(A+B)/2}
=2sin(C/2){cos((A-B)/2)-cos((A+B)/2))}
=2sin(C/2)*2sin(A/2)sin(B/2)
=4sin(A/2)sin(B/2)sin(C/2) >0
∴ cosA+cosB+cosC>1