Question

Довести тотожність:
$sin\alpha +sin\beta *sin(\alpha +\beta )=4sin(\alpha +\beta )/2*sin\alpha /2*sin\beta /2$

Answer 1

Объяснение:

21)

$\displaystyle\\sin\alpha +sin\beta -sin(\alpha +\beta )=(sin\alpha +sin\beta) -sin(\alpha +\beta )=\\\\=2*sin\frac{\alpha +\beta }{2} *cos\frac{\alpha -\beta }{2} -2*sin\frac{\alpha +\beta }{2} *cos\frac{\alpha +\beta }{2} =\\\\=2sin\frac{\alpha +\beta }{2}*(cos\frac{\alpha -\beta }{2} -cos\frac{\alpha +\beta }{2} )=\\\\=2*sin\frac{\alpha +\beta }{2} *(-2sin\frac{\frac{ \alpha -\beta }{2}-\frac{\alpha +\beta }{2} }{2}* sin\frac{\frac{ \alpha -\beta }{2}+\frac{\alpha +\beta }{2} }{2})=$

$\displaystyle\\=4*sin\frac{\alpha +\beta }{2} *(-sin(-\frac{\beta }{2} )*sin\frac{\alpha }{2})=4*sin\frac{\alpha +\beta }{2} *sin\frac{\alpha }{2} *sin\frac{\beta }{2}.$