Cos 2pi/31 cos 8pi/31 cos 16pi/31 cos 32pi/31=1/32 доказать тождество. ответ= верное тождество

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Ответ:

lizabobovic

06.10.2020 21:03

$cos\frac{2\pi }{31}\cdot cos\frac{4\pi }{31}\cdot cos\frac{8\pi }{31}\cdot cos\frac{16\pi }{31}\cdot cos\frac{32\pi }{31}= \frac{1}{32} \\\\\star \; \; 2sina\cdot cosa=sin2a\quad \Rightarrow \quad sina\cdot cosa= \frac{1}{2} \, sin2a\; \; \star \\\\ \frac{1}{2sin\frac{2\pi}{31}} \cdot (2sin\frac{2\pi }{31}\cdot cos\frac{2\pi }{31})\cdot cos\frac{4\pi }{31}\cdot cos\frac{8\pi }{31}\cdot cos\frac{16\pi }{31}\cdot cos\frac{32\pi }{31}=\\\\= \frac{1}{2sin\frac{2\pi}{31}}\cdot \underbrace {sin\frac{4\pi }{31}\cdot cos\frac{4\pi }{31}}\cdot cos\frac{8\pi }{31}\cdot cos\frac{16\pi }{31}\cdot cos\frac{32\pi }{31}=$

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

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

$=\frac{1}{32sin\frac{2\pi}{31}}\cdot sin\frac{2\pi }{31}=\frac{1}{32}$