4 Упростить выражение: 1) sin (α – β) – sin (π/2 – α) · sin (-β); 2) cos² (π – α) – cos² (π/2 – α); 3) 2 sin α sin β + cos (α + β).

1) $$sin(\alpha - \beta) - sin(\frac{\pi}{2} - \alpha) \cdot sin(-\beta) = sin(\alpha - \beta) + cos(\alpha) \cdot sin(\beta) = sin(\alpha)cos(\beta) - cos(\alpha)sin(\beta) + cos(\alpha)sin(\beta) = sin(\alpha)cos(\beta)$$

2) $$cos^2(\pi - \alpha) - cos^2(\frac{\pi}{2} - \alpha) = (-cos\alpha)^2 - (sin\alpha)^2 = cos^2(\alpha) - sin^2(\alpha) = cos(2\alpha)$$

3) $$2 sin(\alpha)sin(\beta) + cos(\alpha + \beta) = 2 sin(\alpha)sin(\beta) + cos(\alpha)cos(\beta) - sin(\alpha)sin(\beta) = sin(\alpha)sin(\beta) + cos(\alpha)cos(\beta) = cos(\alpha - \beta)$$

Ответ:

1. $$sin(\alpha)cos(\beta)$$
2. $$cos(2\alpha)$$
3. $$cos(\alpha - \beta)$$