Упр.20.1 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) 1-cos^2 a; 6) 1/(cos^2 a)-1;
2) sin^2 ?-1; 7) 1-sin^2(a)+cos^2(a)sin^2(a);
3) sin^2(?)+cos^2(?)+1; 8) cos^2(a)+ctg^2(a)-1/sin^2(a);
4) 1-sin^2(3a)-cos^2(3a); 9) sin^2(?)/(1-sin^2(?))·ctg^2(?);
5) cos(a)tg(a); 10) (sin a+cos a)^2+(sin a-cos a)^2.

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

2. $$\sin^2 \beta-1=\sin^2 \beta-(\sin^2 \beta+\cos^2 \beta)=-\cos^2 \beta.$$

3. $$\sin^2 \varphi+\cos^2 \varphi+1=1+1=2.$$

4. $$1-\sin^2 3a-\cos^2 3a=1-(\sin^2 3a+\cos^2 3a)=1-1=0.$$

5. $$\cos a \cdot \tg a=\cos a \cdot \frac{\sin a}{\cos a}=\sin a.$$

6. $$\frac{1}{\cos^2 a}-1=\left(1+\tg^2 a\right)-1=\tg^2 a.$$

7. $$1-\sin^2 a+\ctg^2 a \cdot \sin^2 a=1-\sin^2 a+\frac{\cos^2 a}{\sin^2 a}\cdot \sin^2 a$$
   $$=(\sin^2 a+\cos^2 a)-\sin^2 a+\cos^2 a=2\cos^2 a.$$

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

9. $$\frac{\sin^2 \beta}{1-\sin^2 \beta}\cdot \ctg^2 \beta=\frac{\sin^2 \beta}{\cos^2 \beta}\cdot \frac{\cos^2 \beta}{\sin^2 \beta}=1.$$

10. $$(\sin a+\cos a)^2+(\sin a-\cos a)^2$$
    $$=(\sin^2 a+2\sin a\cos a+\cos^2 a)+(\sin^2 a-2\sin a\cos a+\cos^2 a)$$
    $$=2(\sin^2 a+\cos^2 a)=2.$$

Ответ

1. $$\sin^2 a$$
2. $$-\cos^2 \beta$$
3. $$2$$
4. $$0$$
5. $$\sin a$$
6. $$\tg^2 a$$
7. $$2\cos^2 a$$
8. $$-\sin^2 a$$
9. $$1$$
10. $$2$$