Упр.42.39 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

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

1. $$\sin^3 a+\cos^3 a\cdot \tg a=\sin^3 a+\cos^3 a\cdot \frac{\sin a}{\cos a}$$

   $$=\sin^3 a+\sin a\cdot \cos^2 a=\sin a(\sin^2 a+\cos^2 a)=\sin a.$$

2. $$\cos^2 a-\ctg^2 a\cdot \sin^4 a=\cos^2 a-\frac{\cos^2 a}{\sin^2 a}\cdot \sin^4 a$$

   $$=\cos^2 a-\sin^2 a\cdot \cos^2 a=\cos^2 a(1-\sin^2 a)=\cos^4 a.$$

3. $$\left(\sin^2 a+\tg^2 a\cdot \sin^2 a\right)\ctg a=\sin^2 a(1+\tg^2 a)\ctg a$$

   $$=\sin^2 a\cdot \frac{1}{\cos^2 a}\cdot \frac{\cos a}{\sin a}=\frac{\sin a}{\cos a}=\tg a.$$

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

   $$=\frac{-2\sin a\cos a}{\sin a\cos a-\frac{\cos a}{\sin a}} =\frac{-2\sin a\cos a}{\cos a\left(\sin a-\frac{1}{\sin a}\right)} =\frac{-2\sin a}{\sin a-\frac{1}{\sin a}}$$

   $$=\frac{-2\sin^2 a}{\sin^2 a-1} =\frac{2\sin^2 a}{1-\sin^2 a} =\frac{2\sin^2 a}{\cos^2 a} =2\tg^2 a.$$

Ответ

1) $$\sin a$$; 2) $$\cos^4 a$$; 3) $$\tg a$$; 4) $$2\tg^2 a$$.