Question

Преобразуйте в произведение

Answer 1

$a) \: \: tg \alpha + tg \beta = \frac{sin( \alpha + \beta )}{cos \alpha cos \beta } \\ \\ tg20 + tg70 = \frac{sin(20 + 70)}{cos20cos70} = \frac{sin90}{ \frac{1}{2} (cos50 + cos90)} = \\ = \frac{1}{ \frac{1}{2} cos50} = \frac{2}{cos50}$

$b) \: \: \: {(sin \frac{ \alpha }{2}) }^{2} = \frac{1 - cos \alpha }{2} \\ \\ \: \: 1 - sin \alpha - cos \alpha = (1 - cos \alpha ) -sin \alpha = 2 {(sin \frac{ \alpha }{2} )}^{2} - 2sin \frac{ \alpha }{2} cos \frac{ \alpha }{2} = \\ = 2sin \frac{ \alpha }{2} (sin \frac{ \alpha }{2} - cos \frac{ \alpha }{2} )$


$c) \: \: sin \alpha + sin \beta = 2sin \frac{ \alpha + \beta }{2} cos \frac{ \alpha - \beta }{2} \\ \: \: \: \: \: cos \alpha - cos \beta = - 2sin \frac{ \alpha + \beta }{2}sin \frac{ \alpha - \beta }{2} \\ \\ \: \: \frac{sin25 + sin5}{cos5 - cos25} = \frac{2sin15cos10}{2sin15sin10} = \frac{cos10}{sin10} = ctg10$