Question

cos 2pi/31 cos 8pi/31 cos 16pi/31 cos 32pi/31=1/32 Доказать тождество. Ответ= верное тождество
 

Answer 1

$cosfrac{2pi }{31}cdot cosfrac{4pi }{31}cdot cosfrac{8pi }{31}cdot cosfrac{16pi }{31}cdot cosfrac{32pi }{31}= frac{1}{32} \\star ; ; 2sinacdot cosa=sin2aquad Rightarrow quad sinacdot cosa= frac{1}{2} , sin2a; ; star \\ frac{1}{2sinfrac{2pi}{31}} cdot (2sinfrac{2pi }{31}cdot cosfrac{2pi }{31})cdot cosfrac{4pi }{31}cdot cosfrac{8pi }{31}cdot cosfrac{16pi }{31}cdot cosfrac{32pi }{31}=\\= frac{1}{2sinfrac{2pi}{31}}cdot underbrace {sinfrac{4pi }{31}cdot cosfrac{4pi }{31}}cdot cosfrac{8pi }{31}cdot cosfrac{16pi }{31}cdot cosfrac{32pi }{31}=$

$= frac{1}{2sinfrac{2pi}{31}}cdot frac{1}{2}cdot underbrace {sinfrac{8pi }{31}cdot cosfrac{8pi }{31}}cdot cosfrac{16pi }{31}cdot cosfrac{32pi }{31}=\\=frac{1}{4sinfrac{2pi }{31}}cdot frac{1}{2}cdot underbrace {sinfrac{16pi }{31}cdot cosfrac{16pi }{31}}cdot cosfrac{32pi }{31}=\\= frac{1}{8sinfrac{2pi}{31}}cdot frac{1}{2}cdot underbrace {sinfrac{32pi }{31}cdot cosfrac{32pi }{31}}= frac{1}{16sinfrac{2pi}{31}}cdot frac{1}{2}cdot sinfrac{64pi }{31}=$

$= frac{1}{32sinfrac{2pi}{31}}cdot sinfrac{62pi +2pi }{31}=frac{1}{32sinfrac{2pi}{31}}cdot sin(2pi +frac{2pi }{31})=$

$=frac{1}{32sinfrac{2pi}{31}}cdot sinfrac{2pi }{31}=frac{1}{32}$