cos pi/7 +cos 3pi/7 +cos 5pi/7=1/2

Multiply and divide by 2sin(π/7) 

= 1/2sin(π/7) [ 2sin(π/7)cos(π/7) + 2sin(π/7)cos(3π/7) + 2sin(π/7)cos(5π/7) ] 

Use the following 

sin(2A) = 2sinAcosA 
2sinAcosB = sin(A+B) + sin(A-B) 

= 1/2sin(π/7) [ sin(2π/7) + sin(π/7+3π/7) + sin(π/7-3π/7) + sin(π/7+5π/7) + sin(π/7-5π/7) ] 
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) + sin(-2π/7) + sin(6π/7) + sin(-4π/7) ] 
= 1/2sin(π/7) [ sin(2π/7) + sin(4π/7) - sin(2π/7) + sin(6π/7) - sin(4π/7) ] 
= 1/2sin(π/7) [sin(6π/7)] 
= 1/2sin(π/7) [sin(π - π/7)] 
= [sin(π/7)]/2sin(π/7) 
= 1/2