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) $$\cos^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$

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

Ответ: $$-2\sin2\alpha\sin4\beta$$

2) $$\sin^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$

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

Ответ: $$-\cos2\alpha\cos4\beta$$