Упр.1042 ГДЗ Колягин Ткачёва 10 класс (Алгебра)

1) tgatgb + (tga + tgb)ctg(a + b);
2) (tga — tgb)ctg(a -b) — tgatgb;

1. $$\tg a \tg b + (\tg a + \tg b)\ctg(a+b)$$
   $$=\tg a \tg b + (\tg a + \tg b)\cdot \frac{1-\tg a \tg b}{\tg a+\tg b}$$
   $$=\tg a \tg b + 1 — \tg a \tg b = 1.$$

2. $$ (\tg a-\tg b)\ctg(a-b)-\tg a \tg b $$
   $$=(\tg a-\tg b)\cdot \frac{1+\tg a \tg b}{\tg a-\tg b}-\tg a \tg b$$
   $$=1+\tg a \tg b-\tg a \tg b=1.$$

3. $$\frac{3\ctg^2 15^\circ-1}{3-\ctg^2 15^\circ}$$
   $$=\frac{\ctg^2 30^\circ\cdot \ctg^2 15^\circ-1}{\ctg^2 30^\circ-\ctg^2 15^\circ}$$
   $$=\frac{(\ctg 30^\circ\cdot \ctg 15^\circ-1)(\ctg 30^\circ\cdot \ctg 15^\circ+1)}{(\ctg 30^\circ+\ctg 15^\circ)(\ctg 30^\circ-\ctg 15^\circ)}$$
   $$=\ctg(30^\circ+15^\circ)\cdot \ctg(15^\circ-30^\circ)$$
   $$=\ctg 45^\circ\cdot \ctg(-15^\circ)=-\ctg 15^\circ.$$

4. $$\frac{\tg 50^\circ-\tg 5^\circ-1}{\tg 50^\circ\cdot \tg 5^\circ}$$
   $$=\frac{\tg(50^\circ-5^\circ)(1-\tg 50^\circ\cdot \tg 5^\circ)-1}{\tg 50^\circ\cdot \tg 5^\circ}$$
   $$=\frac{\tg 45^\circ(1-\tg 50^\circ\cdot \tg 5^\circ)-1}{\tg 50^\circ\cdot \tg 5^\circ}$$
   $$=\frac{1-\tg 50^\circ\cdot \tg 5^\circ-1}{\tg 50^\circ\cdot \tg 5^\circ}=-1.$$

Ответ

1) $$1$$;
2) $$1$$;
3) $$-\ctg 15^\circ$$;
4) $$-1$$.