$5.28 \operatorname{tg}\left(x-15^{\circ}\right) \operatorname{ctg}\left(x+15^{\circ}\right)=\frac{1}{3}$.
5.28 Let's move on to sines and cosines and use formula (4.27):

$$
\begin{aligned}
& \frac{2 \sin \left(x-15^{\circ}\right) \cos \left(x+15^{\circ}\right)}{2 \cos \left(x-15^{\circ}\right) \sin \left(x+15^{\circ}\right)}=\frac{1}{3} \\
& \cos \left(x-15^{\circ}\right) \neq 0 ; \sin \left(x+15^{\circ}\right) \neq 0 \\
& \frac{\sin 2 x-\sin 30^{\circ}}{\sin 2 x+\sin 30^{\circ}}=\frac{1}{3} ; \frac{2 \sin 2 x-1}{2 \sin 2 x+1}=\frac{1}{3} ; 2 \sin 2 x+1 \neq 0
\end{aligned}
$$

$6 \sin 2 x-3=2 \sin 2 x+1$

$\sin 2 x=1 ; 2 x=90^{\circ}+360^{\circ} k ; x=45^{\circ}(4 k+1)$.

Answer: $\quad x=45^{\circ}(4 k+1), k \in Z$.