Question

срочно помогите по тригонометрий упростите

Answer 1

$1) sin^{4} frac{23 pi }{12}- cos^{4} frac{13 pi }{12}=sin^4(2 pi - frac{ pi}{12})-cos^4( pi + frac{ pi }{12})=sin^4 frac{ pi }{12}-cos^4 frac{ pi }{12} \ =(sin^2 frac{ pi }{12}-cos^2 frac{ pi }{12})(sin^2 frac{ pi }{12}+cos^2 frac{ pi }{12})=-cos(2* frac{ alpha }{12})=-cos frac{ pi }{6}=- frac{ sqrt{3} }{2};$
$2) tg frac{7 pi }{8}+ctg frac{7 pi}{8}=tg( pi - frac{ pi }{8})+ctg( pi - frac{ pi }{8})=-tg frac{ pi }{8}-ctg frac{ pi }{8}= \=-frac{sin frac{ pi }{4} }{1+cos frac{ pi }{4}}- frac{1+cos frac{ pi }{4}}{sin frac{ pi }{4}}=-( frac{ frac{ sqrt{2} }{2} }{1+ frac{ sqrt{2} }{2}}+ frac{1+ frac{ sqrt{2} }{2} }{ frac{ sqrt{2} }{2}})=-( frac{ frac{1}{2}+ frac{3}{2}+ sqrt{2}}{ frac{ sqrt{2} }{2}+ frac{1}{2}})=$
$=- frac{2(2+ sqrt{2} )}{1+ sqrt{2}}=- frac{2(2+ sqrt{2})(1- sqrt{2})}{1-2}=-2 sqrt{2};$
$3) ctg frac{5 pi }{8}+ctg frac{9 pi }{8}= frac{sin frac{7 pi }{4}}{sin frac{5 pi }{8}sin frac{9 pi }{8}}= frac{- frac{ sqrt{2}}{2}}{ frac{1}{2}(cos frac{ pi }{2}-cos frac{7 pi }{4})}= \ =frac{- frac{ sqrt{2}}{2}}{ frac{1}{2}(0- frac{ sqrt{2}}{2})}= frac{ sqrt{2} }{2}: frac{ sqrt{2} }{4}=2;$

Answer 2

огромное спасибо ))