Problem 5. Since $\left|a+\frac{1}{a}\right| \geqslant 2$ and equality is achieved only when $|a|=1$, the first equation of the system is equivalent to $\operatorname{tg} x= \pm 1, \sin y= \pm 1$. Then from the second equation, $\cos z=0$.
Answer: $\left(\frac{\pi}{4}+\frac{\pi n}{2} ; \frac{\pi}{2}+\pi k ; \frac{\pi}{2}+\pi l\right)$.