Упр.545 ГДЗ Алимов 10-11 класс (Алгебра)

1) 1 — cos а + sin а;
2) 1 — 2 cos а + cos 2а;
3) 1 + sin а — cos а — tg а;
4) 1 + sin а + cos а + tg а.

1. $$1-\cos \alpha+\sin \alpha=\left(\cos^2\frac{\alpha}{2}+\sin^2\frac{\alpha}{2}\right)-\left(\cos^2\frac{\alpha}{2}-\sin^2\frac{\alpha}{2}\right)+2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}$$
   $$=2\sin^2\frac{\alpha}{2}+2\sin\frac{\alpha}{2}\cos\frac{\alpha}{2}=2\sin\frac{\alpha}{2}\left(\sin\frac{\alpha}{2}+\cos\frac{\alpha}{2}\right).$$

2. $$1-2\cos \alpha+\cos 2\alpha=\left(\cos^2\alpha+\sin^2\alpha\right)+\left(\cos^2\alpha-\sin^2\alpha\right)-2\cos\alpha$$
   $$=2\cos^2\alpha-2\cos\alpha=2\cos\alpha(\cos\alpha-1).$$

3. $$1+\sin \alpha-\cos \alpha-\tg \alpha=\frac{\cos\alpha+\sin\alpha\cos\alpha-\cos^2\alpha-\sin\alpha}{\cos\alpha}$$
   $$=\frac{\cos\alpha(1-\cos\alpha)-\sin\alpha(1-\cos\alpha)}{\cos\alpha}$$
   $$=\frac{(1-\cos\alpha)(\cos\alpha-\sin\alpha)}{\cos\alpha}=(1-\cos\alpha)(1-\tg\alpha).$$

4. $$1+\sin \alpha+\cos \alpha+\tg \alpha=\frac{\cos\alpha+\sin\alpha\cos\alpha+\cos^2\alpha+\sin\alpha}{\cos\alpha}$$
   $$=\frac{\cos\alpha(1+\cos\alpha)+\sin\alpha(1+\cos\alpha)}{\cos\alpha}$$
   $$=\frac{(\cos\alpha+\sin\alpha)(1+\cos\alpha)}{\cos\alpha}=(1+\tg\alpha)(1+\cos\alpha).$$

Ответ

1) $$2\sin\frac{\alpha}{2}\left(\sin\frac{\alpha}{2}+\cos\frac{\alpha}{2}\right)$$
2) $$2\cos\alpha(\cos\alpha-1)$$
3) $$\left(1-\cos\alpha\right)\left(1-\tg\alpha\right)$$
4) $$\left(1+\tg\alpha\right)\left(1+\cos\alpha\right)$$