4. Упростите выражение a) $\frac{\cos{\alpha}}{1+\sin{\alpha}} + \tg{\alpha}$, б) $\frac{1}{1+ \tg^2{\alpha}} + \sin^2{\alpha}$

4. a) Упростим выражение: $\frac{\cos{\alpha}}{1+\sin{\alpha}} + \tg{\alpha}$ $\frac{\cos{\alpha}}{1+\sin{\alpha}} + \frac{\sin{\alpha}}{\cos{\alpha}} = \frac{\cos^2{\alpha} + \sin{\alpha}(1+\sin{\alpha})}{(1+\sin{\alpha})\cos{\alpha}} = \frac{\cos^2{\alpha} + \sin{\alpha} + \sin^2{\alpha}}{(1+\sin{\alpha})\cos{\alpha}} = \frac{1 + \sin{\alpha}}{(1+\sin{\alpha})\cos{\alpha}} = \frac{1}{\cos{\alpha}}$ б) Упростим выражение: $\frac{1}{1+ \tg^2{\alpha}} + \sin^2{\alpha}$ Используем тождество: $1 + \tg^2{\alpha} = \frac{1}{\cos^2{\alpha}}$ $\frac{1}{\frac{1}{\cos^2{\alpha}}} + \sin^2{\alpha} = \cos^2{\alpha} + \sin^2{\alpha} = 1$