Rút gọn : cos x + cos (x + 2pi/5) + cos (x + 4pi/5) + cos (x + 6pi/5) + cos (x+ 8pi/5)

Đáp án:

Giải thích các bước giải:

`cos x + cos (x + 2pi/5) + cos (x + 4pi/5) + cos (x + 6pi/5) + cos (x+ 8pi/5)`

 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}\)