Question

Доказать тождество. Помогите пожалуйста )))

Answer 1

$\displaystyle 1) \frac{sin \alpha+tg \alpha }{1+cos \alpha} = \frac{sin\alpha+ \frac{sin\alpha}{cos\alpha} }{1+cos\alpha}= \frac{sin\alpha cos\alpha+sin\alpha}{cos\alpha} \cdot \frac{1}{1+cos\alpha}=$
$\displaystyle = \frac{sin\alpha cos\alpha +sin\alpha}{cos\alpha +cos^2\alpha}= \frac{sin\alpha (cos\alpha +1)}{cos\alpha (1+cos\alpha)}= \frac{sin\alpha}{cos\alpha}= \boxed{tg\alpha}$

$\displaystyle 2) \frac{cos\alpha +ctg\alpha}{1+sin\alpha} = \frac{cos\alpha + \frac{cos\alpha}{sin\alpha} }{1+sin\alpha}= \frac{cos\alpha sin\alpha +cos\alpha}{sin\alpha} \cdot \frac{1}{1+sin\alpha}=$
$\displaystyle = \frac{cos\alpha (sin\alpha +1)}{sin\alpha (1+sin\alpha)}= \frac{cos\alpha}{sin\alpha}= \boxed{ctg\alpha}$