Với mọi a, biểu thức : $A = \cos \alpha {\rm{ + }}\cos \left( {\alpha + \dfrac{\pi }{5}} \right) + ... + \cos \left( {\alpha  + \dfrac{{9\pi }}{5}} \right)$  nhận giá trị bằng :

$A = \cos \alpha {\rm{ + }}\cos \left( {\alpha  + \dfrac{\pi }{5}} \right) + ... + \cos \left( {\alpha  + \dfrac{{9\pi }}{5}} \right)$$A = \left[ {\cos \alpha  + \cos \left( {\alpha  + \dfrac{{9\pi }}{5}} \right)} \right] + ... + \left[ {\cos \left( {\alpha  + \dfrac{{4\pi }}{5}} \right) + \cos \left( {\alpha  + \dfrac{{5\pi }}{5}} \right)} \right]$$A = 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right)\cos \dfrac{{9\pi }}{{10}} + 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right)\cos \dfrac{{7\pi }}{{10}} + ... + 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right)\cos \dfrac{\pi }{{10}}$$A = 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right)\left( {\cos \dfrac{{9\pi }}{{10}} + \cos \dfrac{{7\pi }}{{10}} + \cos \dfrac{{5\pi }}{{10}} + \cos \dfrac{{3\pi }}{{10}} + \cos \dfrac{\pi }{{10}}} \right)$$A = 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right)\left( {2\cos \dfrac{\pi }{2}\cos \dfrac{{2\pi }}{5} + 2\cos \dfrac{\pi }{2}\cos \dfrac{\pi }{5} + \cos \dfrac{\pi }{2}} \right)$$\Leftrightarrow A = 2\cos \left( {\alpha  + \dfrac{{9\pi }}{{10}}} \right).0 = 0.$