Question

упростите выражение а)$frac{(sin alpha -cos alpha ) ^{2}-1 }{ctg alpha -sin alpha cos alpha }$
б)$frac{tg ( pi + alpha )}{1-ctg ^{2}( frac{ pi }{2}+ alpha ) } * frac{1-tg ^{2}( frac{3 pi }{2} - alpha ) }{ctg alpha }$

Answer 1

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

$=-2tg^2 alpha;\\b) frac{tg(pi+ alpha )}{1-ctg^2(frac{pi}{2}+ alpha )}bulletfrac{1-tg^2(frac{3pi}{2}- alpha )}{ctg alpha }=frac{tgalpha }{1-tg^2alpha }bulletfrac{1-ctg^2alpha }{ctg alpha }=frac{tgalpha (1-ctg^2alpha)}{ctg alpha (1-tg^2alpha) }=\\=frac{tgalpha (1-ctg^2alpha)}{ctg alpha (1-tg^2alpha) }=frac{tgalpha-ctgalpha}{ctg alpha-tgalpha }=frac{-(ctg alpha -tg alpha )}{(ctg alpha -tg alpha )}=-1.$