$x=\cos {55}^o,y=\cos {65}^o,z=\cos {175}^{o}thenxy+yx+zx=$

$\begin{array}{l}xy+yz+zx=\cos {55}^o\cos {65}^o+\cos {65}^o\cos {175}^o+\cos {175}^o\cos {55}^o\\ =(1/2)[\cos {120}^o+\cos {10}^o+\cos {240}^o+\cos {110}^o+\cos {230}^o+\cos {120}^o]\\ =(1/2)[2(-1/2)+(-1/2)+2\cos {120}^o\cos {110}^o+\cos {110}^o]\\ =(1/2)[(-3/2)+2(-1/2)\cos {110}^o+\cos {110}^o]=(1/2)[-3/2]=-3/4\end{array}$