rút gọn cos 2pi/5 + cos 4pi/5 +cos6pi/5 + cos 8pi/5 giúp mình với ạ

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1. Giải thích các bước giải:

   Ta có:

   \(\begin{array}{l}
   \cos x + \cos y = 2.\cos \dfrac{{x + y}}{2}.\cos \dfrac{{x – y}}{2}\\
   \cos \dfrac{{2\pi }}{5} + \cos \dfrac{{4\pi }}{5} + \cos \dfrac{{6\pi }}{5} + \cos \dfrac{{8\pi }}{5}\\
   A = \left( {\cos \dfrac{{8\pi }}{5} + \cos \dfrac{{2\pi }}{5}} \right) + \left( {\cos \dfrac{{6\pi }}{5} + \cos \dfrac{{4\pi }}{5}} \right)\\
    = 2.\cos \dfrac{{\dfrac{{8\pi }}{5} + \dfrac{{2\pi }}{5}}}{2}.\cos \dfrac{{\dfrac{{8\pi }}{5} – \dfrac{{2\pi }}{5}}}{2} + 2.\cos \dfrac{{\dfrac{{6\pi }}{5} + \dfrac{{4\pi }}{5}}}{2}.\cos \dfrac{{\dfrac{{6\pi }}{5} – \dfrac{{4\pi }}{5}}}{2}\\
    = 2\cos \pi .\cos \dfrac{{3\pi }}{5} + 2.\cos \pi .\cos \dfrac{\pi }{5}\\
    = 2\cos \pi .\left( {\cos \dfrac{{3\pi }}{5} + \cos \dfrac{\pi }{5}} \right)\\
    =  – 2.\cos \pi .\left( {\cos \left( {\pi  – \dfrac{{3\pi }}{5}} \right) + \cos \left( {\pi  – \dfrac{\pi }{5}} \right)} \right)\\
    =  – 2\cos \pi .\left( {\cos \dfrac{{2\pi }}{5} + \cos \dfrac{{4\pi }}{5}} \right)\\
    = \left( { – 2} \right).\left( { – 1} \right).\left( {\cos \dfrac{{2\pi }}{5} + \cos \dfrac{{4\pi }}{5}} \right)\\
    = 2.\left( {\cos \dfrac{{2\pi }}{5} + \cos \dfrac{{4\pi }}{5}} \right)\\
   \sin x.\cos y = \dfrac{1}{2}\left( {\sin \left( {x + y} \right) + \sin \left( {x – y} \right)} \right)\\
    \Rightarrow A.\sin \dfrac{\pi }{5} = 2.\sin \dfrac{\pi }{5}.\cos \dfrac{{2\pi }}{5} + 2.\sin \dfrac{\pi }{5}.\cos \dfrac{{4\pi }}{5}\\
    \Leftrightarrow A.\sin \dfrac{\pi }{5} = \sin \left( {\dfrac{\pi }{5} + \dfrac{{2\pi }}{5}} \right) + \sin \left( {\dfrac{\pi }{5} – \dfrac{{2\pi }}{5}} \right) + \sin \left( {\dfrac{\pi }{5} + \dfrac{{4\pi }}{5}} \right) + \sin \left( {\dfrac{\pi }{5} – \dfrac{{4\pi }}{5}} \right)\\
    \Leftrightarrow A.\sin \dfrac{\pi }{5} = \sin \dfrac{{3\pi }}{5} + \sin \left( { – \dfrac{\pi }{5}} \right) + \sin \pi  + sin\left( { – \dfrac{{3\pi }}{5}} \right)\\
    \Leftrightarrow A.\sin \dfrac{\pi }{5} = \sin \dfrac{{3\pi }}{5} – \sin \dfrac{\pi }{5} + 0 – \sin \dfrac{{3\pi }}{5}\\
    \Leftrightarrow A.\sin \dfrac{\pi }{5} =  – \sin \dfrac{\pi }{5}\\
    \Leftrightarrow A =  – 1
   \end{array}\)

   Trả lời