Question

довести тотожність................

Answer 1

$1)sin alpha +cos alpha = sqrt2(cos frac{ pi }{4}- alpha )}\sin alpha +cos alpha = sqrt2(cos frac pi 4cos alpha +sin frac pi 4sin alpha )\sin alpha +cos alpha= sqrt{2}( frac{ sqrt{2}}{2}cos alpha + frac{ sqrt{2}}{2}sin alpha )}\sin alpha +cos alpha = frac{( sqrt2)^2}{2}sinalpha +frac{( sqrt2)^2}{2}cos alpha \sin alpha +cos alpha = frac22sin alpha +frac22cos alpha \sin alpha +cos alpha =sin alpha +cos alpha$
$2)sin alpha -cos alpha =- sqrt2(cos frac{ pi }{4}+ alpha )}\sin alpha -cos alpha =- sqrt2(cos frac pi 4cos alpha -sin frac pi 4sin alpha )\sin alpha -cos alpha=- sqrt{2}( frac{ sqrt{2}}{2}cos alpha - frac{ sqrt{2}}{2}sin alpha )}\sin alpha -cos alpha = frac{( sqrt2)^2}{2}sin alpha - frac{( sqrt2)^2}{2}cos alpha \sin alpha -cos alpha = frac22sin alpha - frac22cos alpha \sin alpha -cos alpha =sin alpha -cos alpha$
$3) frac{sin2 alpha +sin6 alpha }{cos2 alpha +cos6 alpha }=tg4 alpha \ frac{2sin( frac{2 alpha +6 alpha}{2} )cos(frac{2 alpha -6 alpha}{2})}{2cos(frac{2 alpha +6 alpha}{2})cos(frac{2 alpha -6 alpha}{2})}=tg4 alpha \ frac{2sin4 alpha }{2cos4 alpha }=tg4 alpha \ frac{sin4 alpha }{cos4 alpha }=tg4 alpha \tg4 alpha =tg4 alpha$
$4) frac{cos2 alpha -cos4 alpha }{cos2 alpha +cos4 alpha }=tg3 alpha tg alpha\ frac{-2sin( frac{2 alpha +4 alpha }{2})sin( frac{2 alpha -4 alpha }{2})}{2cos( frac{2 alpha +4 alpha }{2})cos( frac{2 alpha -4 alpha }{2})}=tg 3alpha tg alpha \ frac{2sin3 alpha sin alpha }{2cos3 alpha cos alpha }=tg3 alpha tg alpha \ frac{sin3 alpha sin alpha }{cos3 alpha cos alpha }=tg3 alpha tg alpha \tg3 alpha tg alpha =tg3 alpha tg alpha$
$5)frac{sin alpha +sin5alpha }{cos alpha +cos5 alpha }=tg3 alpha \ frac{2sin( frac{alpha +5alpha}{2} )cos(frac{alpha -5 alpha}{2})}{2cos(frac{ alpha +5 alpha}{2})cos(frac{ alpha -5alpha}{2})}=tg3 alpha \ frac{2sin3alpha }{2cos3 alpha }=tg3 alpha \ frac{sin3alpha }{cos3 alpha }=tg3 alpha \tg3 alpha =tg3alpha$
$6) frac{sin2 alpha +sin alpha }{sin2 alpha -sin alpha }= frac{tg frac{3 alpha }{2}}{tg frac{ alpha }{2}}\ frac{2sin (frac{2 alpha + alpha }{2})cos (frac{2 alpha - alpha }{2})}{2sin(frac{2 alpha - alpha }{2})cos(frac{2 alpha + alpha }{2})}= frac{tg frac{3 alpha }{2}}{tg frac{ alpha }{2}}\ frac{sin frac{3 alpha }{2}cos frac{ alpha }{2}}{cos frac{3 alpha }{2}sin frac{ alpha }{2}}=frac{tg frac{3 alpha }{2}}{tg frac{ alpha }{2}}$
$frac{tg frac{3 alpha }{2}}{tg frac{ alpha }{2}}=frac{tg frac{3 alpha }{2}}{tg frac{ alpha }{2}}$
УДАЧИ ВАМ ВО ВСЁМ)))!!!