Упр.23.1 ГДЗ Мерзляк 10 класс Углубленный уровень (Алгебра)

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

1. $$\begin{aligned} (1+\tg \alpha)^2+(1-\tg \alpha)^2 &= 1+2\tg \alpha+\tg^2 \alpha+1-2\tg \alpha+\tg^2 \alpha \\ &= 2+2\tg^2 \alpha \\ &= 2(1+\tg^2 \alpha) \\ &= \frac{2}{\cos^2 \alpha}. \end{aligned}$$
2. $$\begin{aligned} \sin^4 \alpha+2\sin^2 \alpha \cos^2 \alpha+\cos^4 \alpha &= (\sin^2 \alpha+\cos^2 \alpha)^2 \\ &= 1^2=1. \end{aligned}$$
3. $$\begin{aligned} \frac{\sin \alpha}{1+\cos \alpha}+\frac{1+\cos \alpha}{\sin \alpha} &= \frac{\sin^2 \alpha+(1+\cos \alpha)^2}{\sin \alpha(1+\cos \alpha)} \\ &= \frac{\sin^2 \alpha+1+2\cos \alpha+\cos^2 \alpha}{\sin \alpha(1+\cos \alpha)} \\ &= \frac{2+2\cos \alpha}{\sin \alpha(1+\cos \alpha)} \\ &= \frac{2(1+\cos \alpha)}{\sin \alpha(1+\cos \alpha)} \\ &= \frac{2}{\sin \alpha}. \end{aligned}$$
4. $$\begin{aligned} \frac{\ctg \alpha}{\tg \alpha+\ctg \alpha} &= \frac{1}{\tg^2 \alpha+1} \\ &= \frac{1}{\frac{1}{\cos^2 \alpha}} \\ &= \cos^2 \alpha. \end{aligned}$$
5. $$\begin{aligned} \frac{1-\ctg \gamma}{1-\tg \gamma} &= \frac{\ctg \gamma-\ctg^2 \gamma}{\ctg \gamma-1} \\ &= \frac{\ctg \gamma(1-\ctg \gamma)}{\ctg \gamma-1} \\ &= -\ctg \gamma. \end{aligned}$$
6. $$\begin{aligned} \cos^4 \alpha-\cos^2 \alpha+\sin^2 \alpha &= \cos^2 \alpha(\cos^2 \alpha-1)+\sin^2 \alpha \\ &= \cos^2 \alpha(-\sin^2 \alpha)+\sin^2 \alpha \\ &= \sin^2 \alpha(1-\cos^2 \alpha) \\ &= \sin^4 \alpha. \end{aligned}$$
7. $$\begin{aligned} \sin^4 \alpha+\sin^2 \alpha \cos^2 \alpha+\cos^2 \alpha &= \sin^2 \alpha(\sin^2 \alpha+\cos^2 \alpha)+\cos^2 \alpha \\ &= \sin^2 \alpha+\cos^2 \alpha \\ &= 1. \end{aligned}$$
8. $$\begin{aligned} \cos(-\alpha)+\cos \alpha \tg^2(-\alpha) &= \cos \alpha+\cos \alpha \tg^2 \alpha \\ &= \cos \alpha(1+\tg^2 \alpha) \\ &= \cos \alpha \cdot \frac{1}{\cos^2 \alpha} \\ &= \frac{1}{\cos \alpha}. \end{aligned}$$

Ответ

1) $$\frac{2}{\cos^2 \alpha}$$; 2) $$1$$; 3) $$\frac{2}{\sin \alpha}$$; 4) $$\cos^2 \alpha$$; 5) $$-\ctg \gamma$$; 6) $$\sin^4 \alpha$$; 7) $$1$$; 8) $$\frac{1}{\cos \alpha}$$.