Solving Trig Problem: Showing Sin(1/9pi) to Sin(4/9pi)=3/16

[mohlam12]

hey,
i have to show that
$$\sin \left( 1/9\,\pi \right) \sin \left( 2/9\,\pi \right) \sin \left( 1/3\,\pi \right) \sin \left( 4/9\,\pi \right) = 3/16$$



i ve tried so many things, and i couldn't get to 3/16 , does anyone have any hints that are going to help me solve problems in that kind!? thanks!

 

[Integral]

It works for me. What have you done?

 

[mohlam12]

Well, i have remplaced sin(2pi/9) by sin(3pi/9 - pi/9) and sin(3pi/9) by sin(4pi/9 - pi/9) and so on, and used the relation sin(a+b)=sina cosb + sinb cos a. so everything will have sin(pi/9) in it, so i can factor with that to get something helpful. but i guess i just messed everything up, and i don't know what relation i can use to get closer the 3/16 or what method i should use

 

[mohlam12]

never mind, i got it