Question

Докажите тождество 645(1)

Answer 1

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

$=\frac{\sqrt{2}sin(\alpha)}{\sqrt{2}cos(\alpha)}=\frac{sin(\alpha)}{cos(\alpha)}=tg(\alpha)$