Упростите тригонометрические выражения

$sin alpha - cos alpha cdot tg alpha= sin alpha -cos alpha cdot frac{sin alpha}{cos alpha}= sin alpha - sin alpha=0$

$sin alpha cdot ctg alpha - 2cos alpha = sin alpha cdot frac{cos alpha}{sin alpha}- 2cos alpha=cos alpha - 2cos alpha = -cos alpha$

$frac{sin^2 alpha}{1-sin^2 alpha}= frac{ frac{1-cos2 alpha}{2} }{1- frac{1-cos 2 alpha}{2}}= frac{ frac{1-cos2 alpha}{2} }{ frac{1+cos 2alpha}{2} }= frac{1-cos 2 alpha}{1+cos 2alpha}= frac{2sin^2 alpha}{2cos^2alpha} =tg^2 alpha$

$frac{1}{1+tg^2alpha}+sin^2 alpha= frac{ 1 }{ frac{1}{cos^2 alpha} }+sin^2 alpha=cos^2 alpha+sin^2 alpha=1$

$frac{1}{1+ctg^2alpha}+sin^2 alpha=frac{ 1 }{ frac{1}{sin^2 alpha} }+sin^2 alpha=sin^2 alpha+sin^2 alpha=2sin^2 alpha= =2 cdot frac{1-cos 2alpha}{2}=1-cos2 alpha$

$(1-cos alpha)(1+cos alpha)=1-cos^2 alpha=1- frac{1+cos2 alpha}{2}= frac{1-cos 2 alpha}{2}=sin^2 alpha$