Question

Прошу помочь в упрощении, в синусах ничего не понимаю... помогите пожалуйста....

Answer 1

$1) frac{1+tg^2alpha}{1+ctg^2alpha}=frac{frac{1}{cos^2alpha}}{frac{1}{sin^2alpha}}=frac{sin^2alpha}{cos^2alpha}=(frac{sin alpha}{cos alpha})^2=tg^2alpha.$

$2) frac{1-(sin alpha +cos alpha )^2}{sin alpha cos alpha -ctg alpha }=frac{1-(sin^2 alpha +2sin alpha cos alpha +cos^2 alpha )}{sin alpha cos alpha -frac{cos alpha }{sin alpha }}=\\=frac{1-((sin^2 alpha +cos^2 alpha) +2sin alpha cos alpha )}{cos alpha(sin alpha -frac{1}{sin alpha })}=frac{1-1 -2sin alpha cos alpha }{cos alpha(sin alpha -frac{1}{sin alpha })}=frac{-2sin alpha }{sin alpha -frac{1}{sin alpha }}=$

$=frac{-2sin alpha }{frac{-(1-sin^2 alpha)}{sin alpha }}=frac{2sin alpha }{frac{cos^2 alpha}{sin alpha }}=frac{2sin alphasin alpha }{cos^2 alpha}=2bulletfrac{sin^2 alpha }{cos^2 alpha }=2bullet(frac{sin alpha }{cos alpha })^2=2tg^2 alpha.$