7. Упростите выражение: a) sin (3 – a) – cos(π + α); 6) tg(π + a) + ctg (-a); B) sin a + (sin a – cos a)²; cos a г) 1-sin a cos a 1+sin a

а) \( sin(\frac{3\pi}{2} - \alpha) - cos(\pi + \alpha) \)

• \( sin(\frac{3\pi}{2} - \alpha) = -cos(\alpha) \).
• \( cos(\pi + \alpha) = -cos(\alpha) \).
• \( -cos(\alpha) - (-cos(\alpha)) = -cos(\alpha) + cos(\alpha) = 0 \).

б) \( tg(\pi + \alpha) + ctg(\frac{\pi}{2} - \alpha) \)

• \( tg(\pi + \alpha) = tg(\alpha) \).
• \( ctg(\frac{\pi}{2} - \alpha) = tg(\alpha) \).
• \( tg(\alpha) + tg(\alpha) = 2tg(\alpha) \).

в) \( sin(\alpha) + (sin(\alpha) - cos(\alpha))^2 \)

• \( (sin(\alpha) - cos(\alpha))^2 = sin^2(\alpha) - 2sin(\alpha)cos(\alpha) + cos^2(\alpha) \).
• \( sin(\alpha) + sin^2(\alpha) - 2sin(\alpha)cos(\alpha) + cos^2(\alpha) = sin(\alpha) + 1 - sin(2\alpha) \).

г) \( \frac{cos(\alpha)}{1 - sin(\alpha)} - \frac{cos(\alpha)}{1 + sin(\alpha)} \)

• Приводим к общему знаменателю:
• \( \frac{cos(\alpha)(1 + sin(\alpha)) - cos(\alpha)(1 - sin(\alpha))}{(1 - sin(\alpha))(1 + sin(\alpha))} \).
• \( \frac{cos(\alpha) + cos(\alpha)sin(\alpha) - cos(\alpha) + cos(\alpha)sin(\alpha)}{1 - sin^2(\alpha)} = \frac{2cos(\alpha)sin(\alpha)}{cos^2(\alpha)} = \frac{2sin(\alpha)}{cos(\alpha)} = 2tg(\alpha) \).