Question

Помогите решить номер 179(если можно,то все буквы)

Answer 1

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

$=frac{cos^2 alpha -sin alpha +sin^2 alpha }{cos alpha (1-sin alpha )}=frac{1-sin alpha }{cos alpha (1-sin alpha )}=frac{1}{cos alpha }\\3)quad 1+cos alpha -frac{sin^2 alpha cos alpha }{1-cos alpha }=frac{(1+cos alpha )(1-cos alpha )-sin^2 alpha cos alpha }{1-cos alpha }=\\=frac{(1-cos^2 alpha )-sin^2 alpha cos alpha }{1-cos alpha }=frac{sin^2 alpha -sin^2 alpha cos alpha }{1-cos alpha }=frac{sin^2 alpha (1-cos alpha )}{1-cos alpha }=sin^2 alpha$

$4)quad 1+sin alpha -frac{cos^2 alpha sin alpha }{1-sin alpha }=frac{(1-sin^2 alpha )-cos^2 alpha sin alpha }{1-sin alpha }=\\=frac{cos^2 alpha -cos^2 alpha sin alpha }{1-sin alpha }=frac{cos^2 alpha (1-sin alpha )}{1-sin alpha }=cos^2 alpha$