Question

2 вариант
Техникум доказать тождество

Answer 1

$\sin(\alpha+\beta)-2\sin\alpha\cos\beta=\sin\alpha\cos\beta+\cos\alpha\sin\beta-2\sin\alpha\cos\beta=\\=\cos\alpha\sin\beta-\sin\alpha\cos\beta=-(\sin\alpha\cos\beta-\cos\alpha\sin\beta)=-\sin(\alpha-\beta)=\sin(\beta-\alpha)\\\\2\sin\alpha\sin\beta+\cos(\alpha+\beta)=2\sin\alpha\sin\beta+\cos\alpha\cos\beta-\sin\alpha\sin\beta=\\=\cos\alpha\cos\beta+\sin\alpha\sin\beta=\cos(\alpha+\beta)=\cos(\beta+\alpha)$

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