Упр.24.21 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) sin(?+?)sin(?-?)=sin^2(?)-sin^2(?);
2) (tg^2(?)-tg^2(?))/(1-tg^2(?)tg^2(?))=tg(?+?)tg(?-?);
3) (tg(?)+tg(?))/tg(?+?)+(tg(?)-tg(?))/tg(?-?)+2tg^2(?)=2/cos^2(?).

1. $$\sin(\alpha+\beta)\sin(\alpha-\beta)=(\sin\alpha\cos\beta+\cos\alpha\sin\beta)(\sin\alpha\cos\beta-\cos\alpha\sin\beta)$$

   $$=\sin^2\alpha\cos^2\beta-\cos^2\alpha\sin^2\beta$$

   $$=\sin^2\alpha(1-\sin^2\beta)-\sin^2\beta(1-\sin^2\alpha)$$

   $$=\sin^2\alpha-\sin^2\alpha\sin^2\beta-\sin^2\beta+\sin^2\alpha\sin^2\beta$$

   $$=\sin^2\alpha-\sin^2\beta.$$

2. $$\frac{\tg^2\alpha-\tg^2\beta}{1-\tg^2\alpha\,\tg^2\beta} =\frac{(\tg\alpha-\tg\beta)(\tg\alpha+\tg\beta)}{(1-\tg\alpha\,\tg\beta)(1+\tg\alpha\,\tg\beta)}$$

   $$=\frac{\tg\alpha-\tg\beta}{1+\tg\alpha\,\tg\beta}\cdot \frac{\tg\alpha+\tg\beta}{1-\tg\alpha\,\tg\beta}$$

   $$=\tg(\alpha-\beta)\tg(\alpha+\beta).$$

3. $$\frac{\tg\alpha+\tg\beta}{\tg(\alpha+\beta)}+\frac{\tg\alpha-\tg\beta}{\tg(\alpha-\beta)}+2\tg^2\alpha$$

   $$=\frac{\tg\alpha+\tg\beta}{\frac{\tg\alpha+\tg\beta}{1-\tg\alpha\,\tg\beta}} +\frac{\tg\alpha-\tg\beta}{\frac{\tg\alpha-\tg\beta}{1+\tg\alpha\,\tg\beta}} +2\tg^2\alpha$$

   $$=1-\tg\alpha\,\tg\beta+1+\tg\alpha\,\tg\beta+2\tg^2\alpha$$

   $$=2+2\tg^2\alpha=2(1+\tg^2\alpha)=\frac{2}{\cos^2\alpha}.$$

Ответ

$$\sin(\alpha+\beta)\sin(\alpha-\beta)=\sin^2\alpha-\sin^2\beta,$$
$$\frac{\tg^2\alpha-\tg^2\beta}{1-\tg^2\alpha\,\tg^2\beta}=\tg(\alpha+\beta)\tg(\alpha-\beta),$$
$$\frac{\tg\alpha+\tg\beta}{\tg(\alpha+\beta)}+\frac{\tg\alpha-\tg\beta}{\tg(\alpha-\beta)}+2\tg^2\alpha=\frac{2}{\cos^2\alpha}.$$