If a=b=c=60º, cosa+cosb+cosc=3*1/2=1.5.
cosc=cos(180-(a+b))=-cos(a+b).
Let a=60-a' and b=60+a' for 0≤a'<60º; cos(0)=1, cos(60)=0.5.
Trig identity: cos(A-B)+cos(A+B)=cosAcos+sinAsinB+cosAcosB-sinAsinB=2cosAcosB.
We have: cos(60-a')+cos(60+a')-cos(120)=2cos(60)cos(a')+0.5=cos(a')+0.5. ≤1.5.
Also, as a+b approaches 0, the expression on the left < 1+1-1 <1. 1 is less than 1.5.
As a+b approaches 180, the expression becomes cosa+cosb+1.
Let a=90-a' and b=90+a', 0<a'<90, cos(90-a')+cos(90+a')+1=2cos90cos(a')+1=1, less than 1.5.