Diagonals of a Cube

Geometry Level 3

If a line makes angles α\alpha , β\beta , γ\gamma and δ\delta with the four body diagonals of a cube and the value of
cos2(α)+cos2(β)+cos2(γ)+cos2(δ)\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) + \cos^2(\delta) can be expressed as pq\frac{p}{q}, where pp and qq are coprime integers, find the value of p+qp + q.

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

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