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

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

Используем тождество $$\sin^2\alpha+\cos^2\alpha=1.$$

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

Используем формулу разности квадратов и тождество $$\sin^2\alpha+\cos^2\alpha=1.$$

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

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

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

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

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

$$=1+\ctg^2\alpha-\ctg^2\alpha-\ctg^4\alpha+\ctg^4\alpha+\ctg^6\alpha=1+\ctg^6\alpha$$

Ответ: 1) $$\cos^2\alpha$$; 2) $$\sin^2\alpha$$; 3) $$2$$; 4) $$\sin^3\alpha$$; 5) $$1$$; 6) $$\sin^2\alpha$$; 7) $$1$$; 8) $$\sin\alpha$$; 9) $$1+\ctg^6\alpha$$.