Упростите выражение: 1) sin a cos 4x + cos a sin 4a; 2) cos 3π cos π-sin 3π sin π; 8 8 8 8 3) sinasin(a + β) + cosa cos(α + β); 4) sin53°cos7° - cos 53°sin(-7°); 5) cos(α + β) + 2sinasinẞ.

\( sin(a + 4\alpha) = sin a cos 4\alpha + cos a sin 4\alpha \)

Ответ: \( sin(a + 4\alpha) \)

\( cos \frac{3\pi}{8} cos \frac{\pi}{8} - sin \frac{3\pi}{8} sin \frac{\pi}{8} = cos(\frac{3\pi}{8} + \frac{\pi}{8}) \)

\( cos(\frac{3\pi}{8} + \frac{\pi}{8}) = cos(\frac{4\pi}{8}) = cos(\frac{\pi}{2}) = 0 \)

Ответ: 0

\( sin a sin(\alpha + \beta) + cos a cos(\alpha + \beta) = cos(a - (\alpha + \beta)) = cos(a - \alpha - \beta) \)

Ответ: \( cos(a - \alpha - \beta) \)

\( sin 53° cos 7° - cos 53° sin(-7°) = sin 53° cos 7° + cos 53° sin(7°) = sin(53° + 7°) \)

\( sin(53° + 7°) = sin(60°) = \frac{\sqrt{3}}{2} \)

Ответ: \( \frac{\sqrt{3}}{2} \)

\( cos(\alpha + \beta) + 2sin \alpha sin \beta = cos \alpha cos \beta - sin \alpha sin \beta + 2 sin \alpha sin \beta = cos \alpha cos \beta + sin \alpha sin \beta = cos(\alpha - \beta) \)

Ответ: \( cos(\alpha - \beta) \)