Докажите тождество sin x + cos x =√2 cos(п/4 - x) sin x - cos x =-√2 cos(п/4 + x)

Решите задачу:

$1)\; \; sinx+cosx=\sqrt2\cdot ({ \frac{1}{\sqrt2}\cdot sinx +\frac{1}{\sqrt2}\cdot c osx )=$

$=[\; sin \frac{\pi }{4}=cos \frac{\pi}{4}= \frac{\sqrt2}{2} =\frac{1}{\sqrt2}\; ] =\\\\=\sqrt2\cdot (sin\frac{\pi}{4}\cdot sinx+cos\frac{\pi}{4}\cdot cosx)=\sqrt2\cdot cos(\frac{\pi}{4}-x)\\\\\\2)\; \; sinx-cosx=\sqrt2\cdot ( \frac{1}{\sqrt2} \cdot sinx- \frac{1}{\sqrt2}\cdot cosx)=\\\\=\sqrt2\cdot (sin \frac{\pi}{4} \cdot sinx-cos\frac{\pi }{4}\cdot cosx)=\\\\=-\sqrt2\cdot (cos\frac{\pi}{4}\cdot cosx-sin\frac{\pi}{4}\cdot sinx)=-\sqrt2\cdot cos(\frac{\pi}{4}+x)$