simplify Sin π/9 sin 2π/9 sin π/3 sin 4π/9

we know sin sin π/3 = √(3)/2

2 Sin π/9 sin 2π/9 sin π/3 sin 4π/9 = Sin π/9 sin π/3 ( 2 sin 2π/9 sin 4π/9)

= Sin π/9 sin π/3 ( cos 2π/9 – 2 cos 2π/ 3) using 2 sin A sin B = cos |(A-B)| - cos (A+B)

= Sin π/9 in π/3 ( Sin π/9 cos 2π/9 + 1/ 2 )

So 4 Sin π/9 sin 2π/9 sin π/3 sin 4π/9 = = Sin π/9 in π/3 ( 2 Sin π/9 cos 2π/9 + 1)

= sin π/3 ( 2Sin π/9 ( 1- 2 sin ^2 π/9) +1 Sin π/9)

= sin π/3 ( 3 Sin π/9 - 4 sin ^3 π/9)
= sin π/3 sin π/3 using sin 3x = 3 sin x – 4 sin ^ 3 x
= sin^2 π/3 = 3/4

Or Sin π/9 sin 2π/9 sin π/3 sin 4π/9 = 3/16