Verified answer

[tex]\displaystyle\bf\\Cos125^\circ+Cos55^\circ=2Cos\frac{125^\circ+55^\circ}{2} \cdot Cos\frac{125^\circ-55^\circ}{2} =\\\\\\=2Cos90^\circ\cdot Cos35^\circ=2\cdot 0\cdot Cos35^\circ=0[/tex]

Второй способ :

[tex]\displaystyle\bf\\Cos125^\circ+Cos55^\circ=Cos(180^\circ-55^\circ)+Cos55^\circ=\\\\\\=-Cos55^\circ+Cos55^\circ=0[/tex]