Trigonometric warping

Geometry Level 4

This question is flagged because I believe that \( \alpha, \beta, \gamma\) should be angles in a triangle.


If \(|\cos{\alpha}\cos{\beta}\cos{\gamma}|=\frac{2}{3}\). Then the maximum value of \(\tan{\alpha}\tan{\beta}+\tan{\beta}\tan{\gamma}+\tan{\gamma}\tan{\alpha}\) is \(X\).

Find \(24X\).

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