4.89 It is known that $\sin \alpha + \sin \beta = 2 \sin (\alpha + \beta), \alpha + \beta \neq 2 \pi n (n \in \mathbb{Z})$. Find $\operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\beta}{2}$.
4.89 We have

$$
2 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}=4 \sin \frac{\alpha+\beta}{2} \cos \frac{\alpha+\beta}{2}
$$

from which
$\frac{\cos \frac{\alpha-\beta}{2}}{\cos \frac{\alpha+\beta}{2}}=2$.

Therefore,

$$
\operatorname{tg} \frac{\alpha}{2} \operatorname{tg} \frac{\beta}{2}=\frac{2 \sin \frac{\alpha}{2} \sin \frac{\beta}{2}}{2 \cos \frac{\alpha}{2} \cos \frac{\beta}{2}}=\frac{\cos \frac{\alpha-\beta}{2}-\cos \frac{\alpha+\beta}{2}}{\cos \frac{\alpha-\beta}{2}+\cos \frac{\alpha+\beta}{2}}=\frac{1}{3}
$$