New Year Quandry

Geometry Level 4

\[ \left | \begin{array}{cccc} 1 & \cos\theta & 0 & 0 \\ \cos\theta & 1 & \cos\alpha & \cos\beta \\ 0 & \cos\alpha & 1 & \cos\gamma \\ 0 & \cos\beta & \cos\gamma & 1 \\ \end{array} \right |=0. \] If \(\theta\) is in the form of \[n\pi+(-1)^n \sin^{-1}\left(\dfrac{\sqrt{\cos^2\alpha+\cos^2\beta-2\cos\alpha \cos\beta \cos\gamma}}{\sin A}\right),\] Find \(A\).

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