Inspired by Daniel Liu

Geometry Level 5

\[\Large f\left( x \right) ={ 8x }^{ 3 }-{ 12x }^{ 2 }+bx+d\]

For variables \(b\) and \(d\) independent of \(x\), we are given the function \(f(x) \) as described above to have roots \( -1\le\alpha ,\beta ,\gamma \le 1\) and that

\[ \alpha +\beta +\gamma =1+4\sin { \frac { \cos ^{ -1 }{ (\alpha ) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\beta) } }{ 2 } } \times \sin { \frac { \cos ^{ -1 }{ (\gamma ) } }{ 2 } } \]

Evaluate \(b+d\).

This problem is original.
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