5. Solve the system

$$
\left\{\begin{array}{c}
4 \cos ^{2} x+\cos ^{2} 5 y=4 \cos x \cdot \cos ^{6} 5 y \\
\lg (x-y)^{2}<2 \lg (2 \pi)-\lg 5-\lg 45
\end{array}\right.
$$
5. Answer: $\left(-\frac{1}{2}, 3\right),\left(3,-\frac{1}{2}\right)$.

If $x y>0$, then $\frac{|x y|}{x y}=1$ and the left side of the inequality is not less than 1, while the right side of the inequality is negative. If $x y<0$, then $\frac{|x y|}{x y}=-1$ and the equation is equivalent to the system $\left\{\begin{array}{l}1-x-y-x y=0, \\ 2 x^{2} y^{2}-2 x^{2} y-2 x y^{2}+2 x y-9=0\end{array} \Leftrightarrow\left\{\begin{array}{l}x+y=1-x y, \\ 2 x y(x y-x-y+1)=9\end{array} \Leftrightarrow\left\{\begin{array}{l}x+y=1-x y=0, \\ 4(x y)^{2}=9 .\end{array}\right.\right.\right.$ When $x y<0$, this system is equivalent to the system $\left\{\begin{array}{l}x y=-\frac{3}{2}, \\ x+y=\frac{5}{2},\end{array}\right.$ which has solutions $\left(-\frac{1}{2}, 3\right)$, $\left(3,-\frac{1}{2}\right)$.