When I'm gone

Geometry Level 4

Let \(0^{\circ}\leq\alpha,\beta,\gamma\leq90^{\circ}\) be angles such that \[\sin\alpha-\cos\beta=\tan\gamma\]\[\sin\beta-\cos\alpha=\cot\gamma\]

and \(\gamma_{1},\gamma_{2},\ldots,\gamma_{i}\) are different possible values of \(\gamma\).Find

\[\displaystyle\sum_{i=1}^{n}\gamma_{i}\]

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