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

1. Докажите тождество:
   1) $$\sin(?+?)\sin(?-?)=\sin^2(?)-\sin^2(?)$$;
   2) $$\frac{\operatorname{tg}^2(?)-\operatorname{tg}^2(?)}{1-\operatorname{tg}^2(?)\operatorname{tg}^2(?)}=\operatorname{tg}(?+?)\operatorname{tg}(?-?)$$;
   3) $$\frac{\operatorname{tg}(?)+\operatorname{tg}(?)}{\operatorname{tg}(?+?)}+\frac{\operatorname{tg}(?)-\operatorname{tg}(?)}{\operatorname{tg}(?-?)}+2\operatorname{tg}^2(?)=\frac{2}{\cos^2(?)}$$.

1) $$\sin(\alpha+\beta)\sin(\alpha-\beta)$$

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

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

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

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

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

Тождество доказано.

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

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

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

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

Тождество доказано.

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

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

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

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

Тождество доказано.