Упр.778 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

2) sin2(a + 2b) + sin2(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) $$-\cos2\alpha\cos4\beta$$.