Объяснение:

21)

[tex]\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})=\\\\[/tex]

[tex]\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}.[/tex]