when attempting to solve the equation, I tried to use product to sums, 2sin(A)sin(B) = cos(A-B) - cos(A+B) and 2sin(A)cos(B) = sin(A+B) + sin(A-B)

so, '

= 1/2sin(4π/9) x (2sin(π/9)sin(2π/9))

= 1/2sin(4π/9) x [cos(π/9-2π/9) - cos(π/9+2π/9)]

= 1/2sin(4π/9) x [cos(-π/9) -cos (3π/9)]

= 1/2sin(4π/9) x [cos(π/9) -cos (π/3)]

= 1/2sin(4π/9)cos(π/9) -1/2sin(4π/9)cos (π/3)

= 1/4(2sin(4π/9)cos(π/9))-1/4(2sin(4π/9)cos (π/3))

= 1/4[ sin(4π/9 + π/9) + sin(4π/9 + π/9) ] -1/4[ sin(4π/9+π/3) + sin(4π/9-π/3) ]

= 1/4sin(5π/9) + 1/4sin(3π/90 -1/4sin(7π/9) -1/4sin(π/9)

From here I see no simplification. However, the answer is still right, just not in exact form. Maybe I should try without making cos(-π/9) = cos(π/9), in any case I'm stuck and I really appreciate anyone who tries to help. Thank you.

[Answer = root(3)/8]