Упр.1306 ГДЗ Алимов 10-11 класс (Алгебра)

1. $$\cos^2(a+2b+\sin^2(a-2b)-1$$
2. $$\sin^2(a+2b+\sin^2(a-2b)-1$$

1)

$$\cos^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$
$$=\cos^2(\alpha+2\beta)-\cos^2(\alpha-2\beta)$$
$$=\bigl(\cos(\alpha+2\beta)-\cos(\alpha-2\beta)\bigr)\bigl(\cos(\alpha+2\beta)+\cos(\alpha-2\beta)\bigr)$$
$$=(-2\sin\alpha\sin2\beta)\cdot(2\cos\alpha\cos2\beta)=-\sin2\alpha\sin4\beta.$$

Ответ: $$-\sin2\alpha\sin4\beta$$

2)

$$\sin^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$
$$=\sin^2(\alpha+2\beta)-\cos^2(\alpha-2\beta)$$
$$=\bigl(\sin(\alpha+2\beta)-\cos(\alpha-2\beta)\bigr)\bigl(\sin(\alpha+2\beta)+\cos(\alpha-2\beta)\bigr)$$
$$=\bigl(\sin\alpha\cos2\beta+\cos\alpha\sin2\beta-\cos\alpha\cos2\beta+\sin\alpha\sin2\beta\bigr)$$
$$\cdot\bigl(\sin\alpha\cos2\beta+\cos\alpha\sin2\beta+\cos\alpha\cos2\beta-\sin\alpha\sin2\beta\bigr)$$
$$=\bigl(\sin\alpha(\cos2\beta+\sin2\beta)-\cos\alpha(\cos2\beta-\sin2\beta)\bigr)$$
$$\cdot\bigl(\sin\alpha(\cos2\beta-\sin2\beta)+\cos\alpha(\cos2\beta+\sin2\beta)\bigr)$$
$$=(\sin\alpha-\cos\alpha)(\sin\alpha+\cos\alpha)(\cos2\beta-\sin2\beta)(\cos2\beta+\sin2\beta)$$
$$=(\sin^2\alpha-\cos^2\alpha)(\cos^2 2\beta-\sin^2 2\beta)=-\cos2\alpha\cdot\cos4\beta.$$

Ответ: $$-\cos2\alpha\cos4\beta$$