Diagonals of a Cube

Geometry Level 3

If a line makes angles \(\alpha\) , \(\beta\) , \(\gamma\) and \(\delta\) with the four body diagonals of a cube and the value of
\[\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) + \cos^2(\delta)\] can be expressed as \(\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, find the value of \(p + q\).

Clarification: Body diagonals of a cube are the diagonals which do not lie along any face of the cube.

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