Tính: cos2pi/31*cos4pi/31*cos8pi/31*cos16pi/31*cos32pi/31 Xin mọi người giúp em ak

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1. Đáp án:

   \[\cos \frac{{2\pi }}{{31}}.\cos \frac{{4\pi }}{{31}}.\cos \frac{{8\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}} = \frac{1}{{32}}\]

   Giải thích các bước giải:

    Ta có:

   \(\begin{array}{l}
   \sin 2x = 2\sin x.\cos x\\
   \cos \frac{{2\pi }}{{31}}.\cos \frac{{4\pi }}{{31}}.\cos \frac{{8\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{2}.\frac{1}{{\sin \frac{{2\pi }}{{31}}}}.\left( {2\sin \frac{{2\pi }}{{31}}.\cos \frac{{2\pi }}{{31}}} \right).\cos \frac{{4\pi }}{{31}}.\cos \frac{{8\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{{2\sin \frac{{2\pi }}{{31}}}}.\sin \frac{{4\pi }}{{31}}.\cos \frac{{4\pi }}{{31}}.\cos \frac{{8\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{{4.\sin \frac{{2\pi }}{{31}}}}.sin\frac{{8\pi }}{{31}}.\cos \frac{{8\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{{8.\sin \frac{{2\pi }}{{31}}}}.sin\frac{{16\pi }}{{31}}.\cos \frac{{16\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{{16.\sin \frac{{2\pi }}{{31}}}}.sin\frac{{32\pi }}{{31}}.\cos \frac{{32\pi }}{{31}}\\
    = \frac{1}{{32.\sin \frac{{2\pi }}{{31}}}}.\sin \frac{{64\pi }}{{31}}\\
    = \frac{1}{{32.\sin \frac{{2\pi }}{{31}}}}.\sin \left( {2\pi  + \frac{{2\pi }}{{31}}} \right)\\
    = \frac{1}{{32.\sin \frac{{2\pi }}{{31}}}}.\sin \frac{{2\pi }}{{31}}\\
    = \frac{1}{{32}}
   \end{array}\)

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