Question

формули подвійного аргументу 1-4 ,помогите хоть началом

Answer 1

$1)quad Formyla:quad tg frac{ alpha }{2} =frac{sin alpha }{1+cos alpha } = frac{1-cos alpha }{sin alpha } quad star \\ frac{sin2 alpha }{1+cos2 alpha } cdot frac{cos alpha }{1+cos alpha } cdot frac{cosfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} =tg alpha cdot frac{cos alpha }{1+cos alpha } cdot frac{cosfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} =$

 $= frac{sin alpha }{cos alpha }cdot frac{cos alpha }{1+cos alpha } cdot frac{cosfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} =$ 

$= frac{sin alpha alpha }{1+cos alpha } cdot frac{cosfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} =tgfrac{ alpha }{2}cdot frac{cosfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} = frac{sinfrac{ alpha }{2}}{1+cosfrac{ alpha }{2}} =tgfrac{ alpha }{4}$

$2)quad frac{1-sin36^circ }{cos36^circ } = frac{1-sin(90^circ -54^circ )}{cos(90^circ -54^circ )} =[, frac{1-cos alpha }{sin alpha }=tgfrac{ alpha }{2}, ]=\\=frac{1-cos54^circ }{sin54^circ}=tgfrac{54^circ }{2}=tg27^circ \\3)quad tg(frac{pi}{4}+frac{ alpha }{2})cdot frac{1-sin alpha }{cos alpha } =tg(frac{pi}{4}+frac{ alpha }{2})cdot frac{1-cos(frac{pi}{2}- alpha )}{sin(frac{pi}{2}- alpha )} =$

$=tg(frac{pi}{4}+frac{ alpha }{2})cdot tg(frac{pi}{4}-frac{ alpha }{2})=tg(frac{pi}{4}+frac{ alpha }{2})cdot ctg(frac{pi}{2}-(frac{pi}{4}-frac{ alpha }{2}))=\\=tg(frac{pi}{4}+frac{ alpha }{2})cdot ctg(frac{pi}{4}+frac{ alpha }{2})=1$

$4)quad frac{tg(frac{5pi}{4}- alpha )(1+sin2 alpha )}{cos(frac{5pi}{2}-2 alpha )} = frac{tg(pi +frac{pi}{4}- alpha )(1+sin2 alpha )}{cos(2pi+frac{pi}{2}-2 alpha )} =\\= frac{tg(frac{pi}{4}- alpha )(1+sin2 alpha )}{sin2 alpha } =frac{tg(frac{pi}{4}- alpha )(1+cos (frac{pi}{2}-2alpha ))}{2sin 2alpha }=$

$= frac{tg(frac{pi}{4}- alpha )cdot 2cos^2(frac{pi}{4}- alpha )}{sin2 alpha } = frac{frac{sin(frac{pi}{4}- alpha )}{cos(frac{pi}{4}- alpha )}cdot 2cos^2(frac{pi}{4}- alpha )}{sin2 alpha } =\\= frac{2sin(frac{pi}{4}- alpha )cdot cos(frac{pi}{4}- alpha )}{sin2 alpha } = frac{sin(frac{pi}{2}-2 alpha )}{sin2 alpha }= frac{cos2 alpha }{sin2 alpha } =ctg2 alpha$