Упр.778 ГДЗ Колягин Ткачёва 11 класс (Алгебра)
- $$\cos^2(a+2b)+\sin^2(a-2b)-1$$
- $$\sin^2(a+2b)+\sin^2(a-2b)-1$$
1) $$\cos^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$
$$\cos^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1 = \cos^2(\alpha+2\beta)-\cos^2(\alpha-2\beta)$$
$$= \bigl(\cos(\alpha+2\beta)-\cos(\alpha-2\beta)\bigr)\bigl(\cos(\alpha+2\beta)+\cos(\alpha-2\beta)\bigr)$$
$$= \bigl(-2\sin\alpha\sin2\beta\bigr)\bigl(2\cos\alpha\cos2\beta\bigr) = -4\sin\alpha\cos\alpha\sin2\beta\cos2\beta$$
$$= -\sin2\alpha\sin4\beta.$$
2) $$\sin^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1$$
$$\sin^2(\alpha+2\beta)+\sin^2(\alpha-2\beta)-1 = \sin^2(\alpha+2\beta)-\cos^2(\alpha-2\beta)$$
$$= \bigl(\sin(\alpha+2\beta)-\cos(\alpha-2\beta)\bigr)\bigl(\sin(\alpha+2\beta)+\cos(\alpha-2\beta)\bigr)$$
$$= \bigl(\sin\alpha\cos2\beta+\cos\alpha\sin2\beta-\cos\alpha\cos2\beta+\sin\alpha\sin2\beta\bigr)$$
$$\cdot \bigl(\sin\alpha\cos2\beta+\cos\alpha\sin2\beta+\cos\alpha\cos2\beta-\sin\alpha\sin2\beta\bigr)$$
$$= (\sin\alpha-\cos\alpha)(\sin\alpha+\cos\alpha)(\cos2\beta-\sin2\beta)(\cos2\beta+\sin2\beta)$$
$$= (\sin^2\alpha-\cos^2\alpha)(\cos^2 2\beta-\sin^2 2\beta) = -\cos2\alpha\cos4\beta.$$
Ответ
1) $$-\sin2\alpha\sin4\beta$$; 2) $$-\cos2\alpha\cos4\beta$$.
