Упр.20.7 ГДЗ Мерзляк 10 класс Базовый уровень (Алгебра)

1) (1+tg a)^2+(1-tg a)^2; 7) (ctg a)/(tg a+ctg a);
2) sin^4(a)+2sin^2(a)cos^2(a)+cos^4(a); 8) (1-ctg ?)/(1-tg ?);
3) (sin a)/(1+cos a)+(sin a)/(1-cos a); 9) cos^4(a)-cos^2(a)+sin^2(a);
4) ctg x+(sin x)/(1+cos a); 10) sin^4(a)+sin^2(a)cos^2(a)+cos^2(a);
5) (sin a)/(1+cos a)+(1+cos a)/(sin a); 11) cos(-a)+cos(a)tg^2(-a);
6) tg(3a)/(tg^2(3a)-1)·(1-ctg^2(3a))/ctg(3a); 12) (1+sin(-?))/cos(-?)-tg(-?).

1. $$\begin{aligned} (1+\tg a)^2+(1-\tg a)^2&=(1+2\tg a+\tg^2 a)+(1-2\tg a+\tg^2 a)\\ &=2+2\tg^2 a=2(1+\tg^2 a)=\frac{2}{\cos^2 a}. \end{aligned}$$
2. $$\begin{aligned} \sin^4 a+2\sin^2 a\cos^2 a+\cos^4 a&=(\sin^2 a+\cos^2 a)^2\\ &=1^2=1. \end{aligned}$$
3. $$\begin{aligned} \frac{\sin a}{1+\cos a}+\frac{\sin a}{1-\cos a} &=\frac{\sin a(1-\cos a)+\sin a(1+\cos a)}{(1+\cos a)(1-\cos a)}\\ &=\frac{2\sin a}{1-\cos^2 a} =\frac{2\sin a}{\sin^2 a} =\frac{2}{\sin a}. \end{aligned}$$
4. $$\begin{aligned} \ctg x+\frac{\sin x}{1+\cos x} &=\frac{\cos x}{\sin x}+\frac{\sin x}{1+\cos x}\\ &=\frac{\cos x(1+\cos x)+\sin^2 x}{\sin x(1+\cos x)}\\ &=\frac{\cos x+\cos^2 x+\sin^2 x}{\sin x(1+\cos x)}\\ &=\frac{\cos x+1}{\sin x(1+\cos x)} =\frac{1}{\sin x}. \end{aligned}$$
5. $$\begin{aligned} \frac{\sin a}{1+\cos a}+\frac{1+\cos a}{\sin a} &=\frac{\sin^2 a+(1+\cos a)^2}{(1+\cos a)\sin a}\\ &=\frac{\sin^2 a+1+2\cos a+\cos^2 a}{(1+\cos a)\sin a}\\ &=\frac{2+2\cos a}{(1+\cos a)\sin a} =\frac{2}{\sin a}. \end{aligned}$$
6. $$\begin{aligned} \frac{\tg 3a}{\tg^2 3a-1}\cdot\frac{1-\ctg^2 3a}{\ctg 3a} &=\frac{\tg 3a}{\tg^2 3a-1}\cdot\frac{(1-\ctg^2 3a)\tg^2 3a}{\ctg 3a\cdot\tg^2 3a}\\ &=\frac{\tg 3a}{\tg^2 3a-1}\cdot\frac{\tg^2 3a-1}{\tg 3a}=1. \end{aligned}$$
7. $$\begin{aligned} \frac{\ctg a}{\tg a+\ctg a} &=\frac{\ctg a\cdot\tg a}{(\tg a+\ctg a)\tg a} =\frac{1}{\tg^2 a+1} =\frac{1}{\frac{1}{\cos^2 a}} =\cos^2 a. \end{aligned}$$
8. $$\begin{aligned} \frac{1-\ctg \gamma}{1-\tg \gamma} &=\frac{(1-\ctg \gamma)\ctg \gamma}{(1-\tg \gamma)\ctg \gamma} =\frac{(1-\ctg \gamma)\ctg \gamma}{\ctg \gamma-1} =-\ctg \gamma. \end{aligned}$$
9. $$\begin{aligned} \cos^4 a-\cos^2 a+\sin^2 a &=\cos^4 a-\cos^2 a+(1-\cos^2 a)\\ &=\cos^4 a-2\cos^2 a+1\\ &=(1-\cos^2 a)^2=\sin^4 a. \end{aligned}$$
10. $$\begin{aligned} \sin^4 a+\sin^2 a\cos^2 a+\cos^2 a &=\sin^4 a+\sin^2 a(1-\sin^2 a)+(1-\sin^2 a)\\ &=\sin^4 a+\sin^2 a-\sin^4 a+1-\sin^2 a=1. \end{aligned}$$
11. $$\begin{aligned} \cos(-a)+\cos a\cdot\tg^2(-a) &=\cos a+\cos a\cdot\tg^2 a\\ &=\cos a(1+\tg^2 a) =\cos a\cdot\frac{1}{\cos^2 a} =\frac{1}{\cos a}. \end{aligned}$$
12. $$\begin{aligned} \frac{1+\sin(-\beta)}{\cos(-\beta)}-\tg(-\beta) &=\frac{1-\sin\beta}{\cos\beta}+\tg\beta\\ &=\frac{1-\sin\beta}{\cos\beta}+\frac{\sin\beta}{\cos\beta} =\frac{1}{\cos\beta}. \end{aligned}$$

Ответ

1. $$\frac{2}{\cos^2 a}$$
2. $$1$$
3. $$\frac{2}{\sin a}$$
4. $$\frac{1}{\sin x}$$
5. $$\frac{2}{\sin a}$$
6. $$1$$
7. $$\cos^2 a$$
8. $$-\ctg \gamma$$
9. $$\sin^4 a$$
10. $$1$$
11. $$\frac{1}{\cos a}$$
12. $$\frac{1}{\cos \beta}$$