In the interval 0<x<=pi/2, y = cos x is greater than or equal to y = -2x/pi + 1.

If, say, A<=B<=C are angles of a triangle in this interval, then

cos A > -2A/pi +1, cos B > -2B/pi +1, and cos C >= -2C/pi + 1

So summuing up the inequalities -->

cos A + cos B + cos C > 1.