1. Упростите: 1) sinx⋅cos3x+cosx⋅sin3x; 2) cos4x⋅cosx+sin4x⋅sinx; 3) sint⋅cos4t−cost⋅sin4t; 4) cos85°⋅cos5°−sin85°⋅sin5°; 5) cos 9°⋅cos 54° + sin 9°⋅sin 54°; 6) √2 sin(a-π/4); 7) (sin10°⋅cos16° + cos10°⋅sin16°)/(cos40°⋅cos14° + sin40°⋅sin14°); 8) (tgy+tg5y)/(1−tgy⋅tg5y); 9) (tg73°−tg28°)/(1+tg73°⋅tg28°). 2. Дано: cos α = 4/5; 0 < α < π/2; cos β = 15/17; 3π/2 < β < 2π. Найдите: cos(α+β).

1. Упрощение выражений:

1. \( \sin x \cos 3x + \cos x \sin 3x = \sin(x + 3x) = \sin 4x \)
2. \( \cos 4x \cos x + \sin 4x \sin x = \cos(4x - x) = \cos 3x \)
3. \( \sin t \cos 4t - \cos t \sin 4t = \sin(t - 4t) = \sin(-3t) = -\sin 3t \)
4. \( \cos 85° \cos 5° - \sin 85° \sin 5° = \cos(85° + 5°) = \cos 90° = 0 \)
5. \( \cos 9° \cos 54° + \sin 9° \sin 54° = \cos(54° - 9°) = \cos 45° = \frac{\sqrt{2}}{2} \)
6. \( \sqrt{2} \sin \left( \alpha - \frac{\pi}{4} \right) = \sqrt{2} \left( \sin \alpha \cos \frac{\pi}{4} - \cos \alpha \sin \frac{\pi}{4} \right) = \sqrt{2} \left( \sin \alpha \frac{\sqrt{2}}{2} - \cos \alpha \frac{\sqrt{2}}{2} \right) = \sin \alpha - \cos \alpha \)
7. \( \frac{\sin 10° \cos 16° + \cos 10° \sin 16°}{\cos 40° \cos 14° + \sin 40° \sin 14°} = \frac{\sin(10° + 16°)}{\cos(40° - 14°)} = \frac{\sin 26°}{\cos 26°} = \tan 26° \)
8. \( \frac{\tan y + \tan 5y}{1 - \tan y \tan 5y} = \tan(y + 5y) = \tan 6y \)
9. \( \frac{\tan 73° - \tan 28°}{1 + \tan 73° \tan 28°} = \tan(73° - 28°) = \tan 45° = 1 \)

2. Нахождение cos(α+β):

Дано: \( \cos \alpha = \frac{4}{5}, 0 < \alpha < \frac{\pi}{2} \) (α — угол I четверти).

Так как \( \alpha \) — угол I четверти, \( \sin \alpha > 0 \).

\( \sin \alpha = \sqrt{1 - \cos^2 \alpha} = \sqrt{1 - \left(\frac{4}{5}\right)^2} = \sqrt{1 - \frac{16}{25}} = \sqrt{\frac{9}{25}} = \frac{3}{5} \).

Дано: \( \cos \beta = \frac{15}{17}, \frac{3\pi}{2} < \beta < 2\pi \) (β — угол IV четверти).

Так как \( \beta \) — угол IV четверти, \( \sin \beta < 0 \).

\( \sin \beta = -\sqrt{1 - \cos^2 \beta} = -\sqrt{1 - \left(\frac{15}{17}\right)^2} = -\sqrt{1 - \frac{225}{289}} = -\sqrt{\frac{64}{289}} = -\frac{8}{17} \).

Используем формулу косинуса суммы: \( \cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta \).

\( \cos(\alpha + \beta) = \left(\frac{4}{5}\right) \cdot \left(\frac{15}{17}\right) - \left(\frac{3}{5}\right) \cdot \left(-\frac{8}{17}\right) = \frac{60}{85} - \left(-\frac{24}{85}\right) = \frac{60}{85} + \frac{24}{85} = \frac{84}{85} \).

Ответ: 1. 1) \( \sin 4x \); 2) \( \cos 3x \); 3) \( -\sin 3t \); 4) 0; 5) \( \frac{\sqrt{2}}{2} \); 6) \( \sin \alpha - \cos \alpha \); 7) \( \tan 26° \); 8) \( \tan 6y \); 9) 1. 2. \( \frac{84}{85} \).