Question

Упростите тригонометрическое выражение:

$\frac{ \sqrt{2}-sin a- cos a }{sina-cosa}$

Answer 1

$\star \; \; sina+cosa=sina+sin(\frac{\pi}{2}-a)=2\cdot sin\frac{\pi }{4}\cdot cos(a-\frac{\pi}{4})=\\=2\sqrt2\cdot cos(a-\frac{\pi}{4})\\\\\star \; \; sina-cosa=sina-sin(\frac{\pi}{2}-a)=2\cdot sin(a-\frac{\pi }{4})\cdot cos\frac{\pi}{4}=\\=2\sqrt2\cdot sin(a-\frac{\pi}{4})\\\\\frac{\sqrt2-sina-cosa}{sina+cosa}=\frac{\sqrt2-(sina+cosa)}{sina-cosa}=\frac{\sqrt2-2\sqrt2\cdot cos(a-\frac{\pi}{4})}{2\sqrt2\cdot sin(a-\frac{\pi}{4})}=\\\\=\frac{1}{2\, sin(a-\frac{\pi}{4})}-ctg(a-\frac{\pi}{4})=\frac{1}{2}\cdot cosec(a-\frac{\pi}{4})-ctg(a-\frac{\pi}{4})$