Упр.7.62 ГДЗ Погорелов 7-9 класс (Геометрия)
- 7-62. 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) $$1-\sin^2\alpha=\cos^2\alpha$$
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) $$\tg^2\alpha-\sin^2\alpha\cdot\tg^2\alpha=\tg^2\alpha\left(1-\sin^2\alpha\right)=\tg^2\alpha\cdot\cos^2\alpha=\frac{\sin^2\alpha}{\cos^2\alpha}\cdot\cos^2\alpha=\sin^2\alpha$$
7) $$\sin^2\alpha+\ctg^2\alpha\cdot\sin^2\alpha=\sin^2\alpha\left(1+\ctg^2\alpha\right)=\sin^2\alpha\cdot\frac{1}{\sin^2\alpha}=1$$
8) $$\tg\alpha\left(2\cos\alpha+\sin\alpha-1\right)=\tg\alpha\left(\cos\alpha+\left(\cos^2\alpha+\sin^2\alpha\right)-1\right)$$
$$=\tg\alpha\left(\cos\alpha+\cos^2\alpha\right)=\tg\alpha\cdot\cos\alpha\left(1+\cos\alpha\right)=\sin\alpha\left(1+\cos\alpha\right)$$
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) $$\sin\alpha(1+\cos\alpha)$$; 9) $$1+\ctg^6\alpha$$.
