22.4. Упростите выражение: 9) $$\frac{\sin \beta}{1 + \cos \beta} + ctg \frac{\beta}{2}$$

9) Используем формулы: $$\sin \beta = 2\sin \frac{\beta}{2} \cos \frac{\beta}{2}$$ и $$1 + \cos \beta = 2\cos^2 \frac{\beta}{2}$$. Тогда, $$\frac{\sin \beta}{1 + \cos \beta} = \frac{2\sin \frac{\beta}{2} \cos \frac{\beta}{2}}{2\cos^2 \frac{\beta}{2}} = \frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}} = tg \frac{\beta}{2}$$. $$\frac{\sin \beta}{1 + \cos \beta} + ctg \frac{\beta}{2} = tg \frac{\beta}{2} + ctg \frac{\beta}{2} = \frac{\sin \frac{\beta}{2}}{\cos \frac{\beta}{2}} + \frac{\cos \frac{\beta}{2}}{\sin \frac{\beta}{2}} = \frac{\sin^2 \frac{\beta}{2} + \cos^2 \frac{\beta}{2}}{\sin \frac{\beta}{2} \cos \frac{\beta}{2}} = \frac{1}{\sin \frac{\beta}{2} \cos \frac{\beta}{2}} = \frac{2}{2\sin \frac{\beta}{2} \cos \frac{\beta}{2}} = \frac{2}{\sin \beta}$$