Question

prove without using calculator cos2π/15.cos8π/15.cos14π/15=1/16



pls help.... show proper working

Answer 1

Solution

====================== 
cos(2π/15) cos(4π/15) cos(8π/15) cos(14π/15) 

= cos(2π/15) cos(4π/15) cos(8π/15) cos(π - π/15)
 
= cos(2π/15) cos(4π/15) cos(8π/15) * -cos(π/15) 

= -cos(π/15) cos(2π/15) cos(4π/15) cos(8π/15) 

= -16sin(π/15) cos(π/15) cos(2π/15) cos(4π/15)
cos(8π/15) / [ 16 sin(π/15) ] 

= -8 * [ 2sin(π/15) cos(π/15) ] cos(2π/15)
cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ] 

= -8 sin(2π/15) cos(2π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ] 

= -4 * [ 2 sin(2π/15) cos(2π/15) ] cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ] 

= -4 sin(4π/15) cos(4π/15) cos(8π/15) / [ 16 sin(π/15) ] 

= -2 * [2 sin(4π/15) cos(4π/15) ] cos(8π/15) / [ 16 sin(π/15) ] 

= -2 sin(8π/15) cos(8π/15) / [ 16 sin(π/15) ] 

= -sin(16π/15) / [ 16 sin(π/15) ] 

= -sin(π + π/15) / [ 16 sin(π/15) ] 

= - * -sin(π/15) / [ 16 sin(π/15) ] 

= 1/16