Question

cosπ/11+cos3π/11+cos5π/11+cos7π/11+cos9π/11

Answer 1

Multiplying the given expression by Sin(5π/11), 

Sin(5π/11)Cos(π/11) + Sin(5π/11)Cos(3π/11) + Sin(5π/11)Cos(5π/11) + 
Sin(5π/11)Cos(7π/11) + Sin(5π/11)Cos(9π/11)

Use the formula SinC SinD = (1/2)[Sin(C + D) + Sin(C - D)

= (1/2) [ {Sin(6π/11) + Sin(4π/11)} + {Sin(8π/11) + Sin(2π/11)} 
  + {Sin(10π/11) + Sin(0)} + {Sin(12π/11) + Sin(-2π/11)}
 + {Sin(14π/11) + Sin(-4π/11)]
=  (1/2) [ {Sin(6π/11) + Sin(4π/11)} + {Sin(8π/11) + Sin(2π/11)} 
  + {Sin(10π/11)} + {Sin(-10π/11) + Sin(-2π/11)}
 + {Sin(-8π/11) + Sin(-4π/11)]

=   (1/2) [ Sin(6π/11) + {Sin(4π/11) + Sin(-4π/11)}+ {Sin(8π/11) + Sin(-8π/11)}
   + {Sin(2π/11) + Sin(-2π/11)} + {Sin(10π/11) + Sin(-10π/11)}] 
  
= (1/2) x [Sin(6π/11)] 
AS Sin A = Sin(π - A), 
= (1/2) x {Sin(5π/11)} -------------------------(1)

Since we had multiplied the given expression by Sin(5π/11), we divide by the same now,

The original given expression = [(1/2) x {Sin(5π/1)}] / {Sin(5π/11)}
                                          = 1/2

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