Упр.1309 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\frac{\sqrt{2}-\cos x-\sin x}{\sin x-\cos x} = \frac{\sqrt{2}-\left(\sin\left(\frac{\pi}{2}-x\right)+\sin x\right)}{\sin x-\sin\left(\frac{\pi}{2}-x\right)}$$
$$= \frac{\sqrt{2}-2\sin\frac{\pi}{4}\cdot \cos\left(\frac{\pi}{4}-x\right)}{2\sin\left(x-\frac{\pi}{4}\right)\cdot \cos\frac{\pi}{4}} = \frac{\sqrt{2}-\sqrt{2}\cos\left(\frac{\pi}{4}-x\right)}{\sqrt{2}\sin\left(x-\frac{\pi}{4}\right)}$$
$$= \frac{1-\cos\left(x-\frac{\pi}{4}\right)}{\sin\left(x-\frac{\pi}{4}\right)}.$$

Ответ: $$\frac{1-\cos\left(x-\frac{\pi}{4}\right)}{\sin\left(x-\frac{\pi}{4}\right)}$$

2)

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

Ответ: $$1+\frac{1}{\cos x}$$