Упр.551 ГДЗ Алимов 10-11 класс (Алгебра)

1)

$$\frac{\sin\left(\frac{\pi}{4}+a\right)-\cos\left(\frac{\pi}{4}+a\right)}{\sin\left(\frac{\pi}{4}+a\right)+\cos\left(\frac{\pi}{4}+a\right)} = \frac{\sin\frac{\pi}{4}\cos a+\cos\frac{\pi}{4}\sin a-\cos\frac{\pi}{4}\cos a+\sin\frac{\pi}{4}\sin a}{\sin\frac{\pi}{4}\cos a+\cos\frac{\pi}{4}\sin a+\cos\frac{\pi}{4}\cos a-\sin\frac{\pi}{4}\sin a}$$
$$= \frac{\frac{\sqrt2}{2}\cos a+\frac{\sqrt2}{2}\sin a-\frac{\sqrt2}{2}\cos a+\frac{\sqrt2}{2}\sin a}{\frac{\sqrt2}{2}\cos a+\frac{\sqrt2}{2}\sin a+\frac{\sqrt2}{2}\cos a-\frac{\sqrt2}{2}\sin a} = \frac{\sqrt2\sin a}{\sqrt2\cos a} = \tg a$$

Ответ: $$\tg a$$

2)

$$\frac{\sin\left(\frac{\pi}{4}-a\right)-\cos\left(\frac{\pi}{4}-a\right)}{\sin\left(\frac{\pi}{4}-a\right)+\cos\left(\frac{\pi}{4}-a\right)} = \frac{\sin\frac{\pi}{4}\cos a-\cos\frac{\pi}{4}\sin a-\cos\frac{\pi}{4}\cos a-\sin\frac{\pi}{4}\sin a}{\sin\frac{\pi}{4}\cos a-\cos\frac{\pi}{4}\sin a+\cos\frac{\pi}{4}\cos a+\sin\frac{\pi}{4}\sin a}$$
$$= \frac{\frac{\sqrt2}{2}\cos a-\frac{\sqrt2}{2}\sin a-\frac{\sqrt2}{2}\cos a-\frac{\sqrt2}{2}\sin a}{\frac{\sqrt2}{2}\cos a-\frac{\sqrt2}{2}\sin a+\frac{\sqrt2}{2}\cos a+\frac{\sqrt2}{2}\sin a} = \frac{-\sqrt2\sin a}{\sqrt2\cos a} = -\ctg a$$

Ответ: $$-\ctg a$$