Упр.7.59 ГДЗ Никольский 10 класс (Алгебра)

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

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

   Так как $$\sin^2\alpha+\cos^2\alpha=1,$$ то
   $$\sin^4\alpha-\cos^4\alpha=\sin^2\alpha-\cos^2\alpha.$$

3. $$\sin^4\alpha-\cos^4\alpha-\sin^2\alpha+\cos^2\alpha=0.$$

   Действительно,
   $$\sin^4\alpha-\cos^4\alpha-\sin^2\alpha+\cos^2\alpha=(\sin^2\alpha-\cos^2\alpha)-\sin^2\alpha+\cos^2\alpha=0.$$

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

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

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

Ответ

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