Пожалуйста, помогите доказать тождество!!!!

$\displaystyle \frac{(1+tg \alpha )\sin(\frac{\pi}{4}- \alpha )}{1-tg \alpha } = \frac{ (\cos \alpha +\sin \alpha )\sin(\frac{\pi}{4}- \alpha )}{\cos \alpha -\sin \alpha } =\\ \\ \\ = \frac{\sqrt{2}\sin( \alpha +\arcsin \frac{1}{\sqrt{2}})\sin(\frac{\pi}{4}- \alpha ) }{\sqrt{2}\sin(\arcsin \frac{1}{\sqrt{2}} - \alpha )} = \frac{\sin( \alpha +\frac{\pi}{4})\sin(\frac{\pi}{4}- \alpha )}{\sin(\frac{\pi}{4}- \alpha )} =\sin(\alpha+\frac{\pi}{4})$

P.S. $a\sin x\pm b\cos x= \sqrt{a^2+b^2}\sin(x\pm \arcsin \frac{b}{ \sqrt{a^2+b^2} })$