4. Доказать тождество: $$\frac{1 + \operatorname{tg} \alpha}{1 + \operatorname{ctg} \alpha} = \operatorname{tg} \alpha$$.

$$\frac{1 + \operatorname{tg} \alpha}{1 + \operatorname{ctg} \alpha} = \frac{1 + \frac{\sin \alpha}{\cos \alpha}}{1 + \frac{\cos \alpha}{\sin \alpha}} = \frac{\frac{\cos \alpha + \sin \alpha}{\cos \alpha}}{\frac{\sin \alpha + \cos \alpha}{\sin \alpha}} = \frac{\cos \alpha + \sin \alpha}{\cos \alpha} \cdot \frac{\sin \alpha}{\sin \alpha + \cos \alpha} = \frac{\sin \alpha}{\cos \alpha} = \operatorname{tg} \alpha$$

Тождество доказано