$\mathrm{x}=\cos {55}^{\circ },\mathrm{y}=\cos {65}^{\circ },\mathrm{z}=\cos {175}^{\circ }\Rightarrow \mathrm{xy}+\mathrm{yz}+\mathrm{zx}=$_______

$$\begin{gathered} x+y+z=\cos 55°+\cos 65°+\cos 175°=0 \\ {x}^2+y^2+z^2={\cos }^{2}55°+{\cos }^{2}65°+{\cos }^{2}175°=\frac{3}{2} \\ \Rightarrow {\left(x+y+z\right)}^2=0 \\ \Rightarrow x^2+y^2+z^2+2(xy+yz+zx)=0 \\ \Rightarrow \frac{3}{2}+2(xy+yz+zx)=0 \\ \Rightarrow xy+yz+zx=\frac{-3}{4} \end{gathered}$$