5.60 Solve the system of equations

$$
\left\{\begin{array}{l}
\sin x \sin y=0.75 \\
\operatorname{tg} x \operatorname{tg} y=3
\end{array}\right.
$$
5.60 We have

$$
\left\{\begin{array}{l}
\sin x \sin y=0.75 \\
\frac{\sin x}{\cos x} \cdot \frac{\sin y}{\cos y}=3
\end{array}\right.
$$

From the second equation, we get $\frac{0.75}{\cos x \cos y}=3$, hence $\cos x \cos y=0.25$. Then we arrive at the system

$\left\{\begin{array}{l}\sin x \sin y=0.75 \\ \cos x \cos y=0.25\end{array}\right.$

By adding the equations and subtracting the first equation from the second, we obtain

$$
\begin{aligned}
& \left\{\begin{array}{l}
\sin x \sin y+\cos x \cos y=1 \\
\cos x \cos y-\sin x \sin y=-0.5
\end{array}\right. \\
& \left\{\begin{array}{l}
\cos (x-y)=1 \\
\cos (x+y)=-0.5
\end{array}\right. \\
& \left\{\begin{array}{l}
x-y=2 \pi k \\
x+y= \pm \frac{2 \pi}{3}+2 \pi n
\end{array}\right.
\end{aligned}
$$

Answer: $x= \pm \frac{\pi}{3}+\pi(k+n), y= \pm \frac{\pi}{3}+\pi(n-k), n, k \in \boldsymbol{Z}$.