Question

Cos p/11+cos 3p/11+cos 5p/11 + cos 7p/11+cos 9p/11

Answer 1

   
 Домножим каждое слагаемое  на $sin\frac{\pi}{11}$  
  $cos\frac{\pi}{11}*sin\frac{\pi}{11} = 0.5*sin\frac{2\pi}{11} \\ cos\frac{\pi*3}{11}*sin\frac{3\pi}{11} = 0.5* ( cos\frac{3\pi}{22} - sin\frac{2\pi}{11}) \\ cos\frac{\pi*5}{11}*sin\frac{7\pi}{11 } = 0.5* ( cos\frac{\pi}{22} - cos\frac{3\pi}{22} ) \\\ cos\frac{7\pi*3}{11}*sin\frac{3\pi}{11} = 0.5*(cos\frac{5\pi}{22} - cos\frac{\pi}{22} ) \\ .....$ 
 суммирую и деля 
 $0.5*\frac{sin\frac{\pi}{11}}{sin\frac{\pi}{11}} = 0.5$ 
      

Answer 2

(сosπ/11+cos9π/11)+(cos(3π/11+cos7π/11)+cos5π/11=
=2cos5π/11cos4π/11+2cos5π/11cos2π/11)+cos5π/11=
=2cos5π/11(cos4π/11+cos2π/11)+cos5π/11=2cos5π/11*2cos3π/11*cosπ/11+cos5π/11=
=cos5π/11(4cos3π/11*cosπ/11+1)