Since 0 ≤ a, b, c ≤ 1 , there exist x, y, z such that  0 ≤ x, y, z ≤ π / 2 and sin² x = a, sin² y = b and sin² z = c.

In new variables our inequality has the following form :

sinx cosy cosz  +  siny cosx cosz  +  sinz cosx cosy  ≤  1  +  sinx siny sinz              (*)

Let us prove (*) :

Since sin(x + y + z) ≤ 1 ,

sin(x + y + z)  =  sinz cos(x+y)  +  cosz sin (x+y)  =  sinz ( cosx cosy  -  sinx siny)  +  cosz (sinx cosy  +  cosx siny)

                      =  sinx cosy cosz  +  siny cosx cosz  +  sinz cosx cosy  -  sinx siny sinz    ≤ 1.

The inequality (*) is proved.