Упр.56 Повторение ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

1) (sin^2(?)-1)/(cos^2(?)-1)+tg(?)ctg(?);
2) (tg(?)cos(?))^2+(ctg(?)sin(?))^2;
3) sin^2(?)+sin^2(?)cos^2(?)+cos^4(?);
4) (tg(?)+ctg(?))^2-(tg(?)-ctg(?))^2.

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

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

3. $$\sin^2\alpha+\sin^2\alpha\cos^2\alpha+\cos^4\alpha$$
   $$=\sin^2\alpha+\cos^2\alpha(\sin^2\alpha+\cos^2\alpha)$$
   $$=\sin^2\alpha+\cos^2\alpha$$
   $$=1.$$

4. $$(\tg\beta+\ctg\beta)^2-(\tg\beta-\ctg\beta)^2$$
   $$=\left(\tg^2\beta+2+\ctg^2\beta\right)-\left(\tg^2\beta-2+\ctg^2\beta\right)$$
   $$=4.$$

Ответ

1) $$\frac{1}{\sin^2\alpha}$$; 2) $$1$$; 3) $$1$$; 4) $$4$$.