Упр.575 ГДЗ Никольский Потапов 9 класс (Алгебра)

1. а) $$1-\sin^2\alpha-\cos^2\alpha$$
2. б) $$\sin^4\alpha-\cos^4\alpha$$
3. в) $$\sin^4\alpha-\cos^4\alpha-\sin^2\alpha+\cos^2\alpha$$
4. г) $$(\sin\alpha+\cos\alpha)^2+(\sin\alpha-\cos\alpha)^2$$

Воспользуемся основным тригонометрическим тождеством:

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

1. $$1-\sin^2 \alpha-\cos^2 \alpha$$

   $$1-\sin^2 \alpha-\cos^2 \alpha=(\sin^2 \alpha+\cos^2 \alpha)-\sin^2 \alpha-\cos^2 \alpha=0.$$

2. $$\sin^4 \alpha-\cos^4 \alpha$$

   $$\sin^4 \alpha-\cos^4 \alpha=(\sin^2 \alpha-\cos^2 \alpha)(\sin^2 \alpha+\cos^2 \alpha)=\sin^2 \alpha-\cos^2 \alpha.$$

3. $$\sin^4 \alpha-\cos^4 \alpha-\sin^2 \alpha+\cos^2 \alpha$$

   $$\sin^4 \alpha-\cos^4 \alpha-\sin^2 \alpha+\cos^2 \alpha =(\sin^2 \alpha-\cos^2 \alpha)(\sin^2 \alpha+\cos^2 \alpha)-\sin^2 \alpha+\cos^2 \alpha$$

   $$=\sin^2 \alpha-\cos^2 \alpha-\sin^2 \alpha+\cos^2 \alpha=0.$$

4. $$\left(\sin \alpha+\cos \alpha\right)^2+\left(\sin \alpha-\cos \alpha\right)^2$$

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

   $$=2\sin^2 \alpha+2\cos^2 \alpha=2(\sin^2 \alpha+\cos^2 \alpha)=2.$$

Ответ

а) $$0$$; б) $$\sin^2 \alpha-\cos^2 \alpha$$; в) $$0$$; г) $$2$$.