Упр.7.62 ГДЗ Погорелов 7-9 класс (Геометрия)

Упростите выражения:

1. $$1-\sin^2\alpha$$
2. $$(1-\cos\alpha)(1+\cos\alpha)$$
3. $$1+\sin^2\alpha+\cos^2\alpha$$
4. $$\sin\alpha-\sin\alpha\cos^2\alpha$$
5. $$\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha\cos^2\alpha$$
6. $$\tg^2\alpha-\sin^2\alpha\tg^2\alpha$$
7. $$\sin^2\alpha+\ctg^2\alpha\sin^2\alpha$$
8. $$\tg\alpha(2\cos\alpha+\sin\alpha-1)$$
9. $$\frac{1-\ctg^2\alpha+\ctg^4\alpha}{\sin^2\alpha}$$

1) $$1-\sin^2\alpha=\cos^2\alpha$$

2) $$\left(1-\cos\alpha\right)\left(1+\cos\alpha\right)=1-\cos^2\alpha=\sin^2\alpha$$

3) $$1+\sin^2\alpha+\cos^2\alpha=1+\left(\sin^2\alpha+\cos^2\alpha\right)=2$$

4) $$\sin\alpha-\sin\alpha\cos^2\alpha=\sin\alpha\left(1-\cos^2\alpha\right)=\sin\alpha\cdot\sin^2\alpha=\sin^3\alpha$$

5) $$\sin^4\alpha+\cos^4\alpha+2\sin^2\alpha\cos^2\alpha=\left(\sin^2\alpha+\cos^2\alpha\right)^2=1$$

6) $$\tg^2\alpha-\sin^2\alpha\,\tg^2\alpha=\tg^2\alpha\left(1-\sin^2\alpha\right)=\tg^2\alpha\cos^2\alpha=\sin^2\alpha$$

7) $$\sin^2\alpha+\ctg^2\alpha\,\sin^2\alpha=\sin^2\alpha\left(1+\ctg^2\alpha\right)=\sin^2\alpha\cdot\frac{1}{\sin^2\alpha}=1$$

8) $$\tg\alpha\left(2\cos^2\alpha+\sin^2\alpha-1\right)=\tg\alpha\left(\cos^2\alpha+\left(\cos^2\alpha+\sin^2\alpha\right)-1\right)=\tg\alpha\cos^2\alpha=\sin\alpha\cos\alpha$$

9) $$\frac{1-\ctg^2\alpha+\ctg^4\alpha}{\sin^2\alpha}=\left(1-\ctg^2\alpha+\ctg^4\alpha\right)\left(1+\ctg^2\alpha\right)=1+\ctg^6\alpha$$