I got it! I got it!
Use the identity
Substitute into that. Thus you’ll find that are the roots of the equation
Obviously ; therefore they are the roots of .
Now , etc. Hence the roots of the equation
are . All you need from the sextic equation are the leading coefficient, which is 64, and the constant term, which is . Now multiply all the six roots together and you have your answer!
Oh, I am so happy!
Consider the equation, . The solutions are where .
Thus, we can factorize, .
Let and we get, . This implies, .
Note, .
Which means, .
Thus, we established that, .
In the product,
Look at the sines from .
Use the fact that , , ... , .
This means, we can write this product as,
.
This means, (note ),
.
Now, use fact that, .
Apply the above identity to the opposite ends and work inside-out.
.
Thus,
.
Generalize! If is odd then: .