Question

тригонометрия ,помогите пожалуйста (хотя бы 2 и 4)

Answer 1

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

$*******************************************$

$frac{sin^2alpha}{cos alpha} - frac{cos^2 alpha}{sin alpha}$, если  $sin alpha-cos alpha= frac{1}{2}$

$sinalpha-cosalpha= frac{1}{2}\\ (sinalpha-cosalpha)^2= frac{1}{4}\\ sin^2alpha-2sinalphacosalpha+cos^2alpha= frac{1}{4}\\ 1-2sinalphacosalpha= frac{1}{4}\\ -2sinalphacosalpha=- frac{3}{4} \\ sinalphacosalpha= frac{3}{4} cdot frac{1}{2}\\ sinalphacosalpha = frac{3}{8}$

$frac{sin^2alpha}{cos alpha} - frac{cos^2 alpha}{sin alpha}= frac{(sin alpha-cosalpha)(sin^2 alpha+sinalphacdot cosalpha+cos^2 alpha)}{sinalphacdotcosalpha} =\\ frac{ frac{1}{2}(sin^2 alpha+sinalphacdot cosalpha+cos^2 alpha) }{sinalphacdotcosalpha}= frac{ frac{1}{2}(1+ frac{3}{8}) }{ frac{3}{8} }= frac{1}{2}cdot frac{11}{8} cdot frac{8}{3}= frac{11}{6}$