It's not Pythagoras

Geometry Level pending

Let \(a\), \(b\), and \(c\) be the three sides of a triangle, and let \(\alpha\), \(\beta\), and \(\gamma\) be the angles opposite them respectively. If \(a^2+b^2=1989c^2\), find

\[\large \frac{\color{blue}{\tan \alpha \ \tan \beta}}{\color{magenta}{\tan \alpha \ \tan \gamma}+\color{purple}{\tan \beta \ \tan \gamma}}\]

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