Tan-ning the Sine

Geometry Level 3

\[\begin{eqnarray} &&\tan^{2}\alpha.\tan^{2}\beta + \tan^{2}\alpha.\tan^{2}\gamma \\ && + \tan^{2}\beta.\tan^{2}\gamma + 2\tan^{2}\alpha.\tan^{2}\beta.\tan^{2}\gamma = 1\end{eqnarray}\]

If \(\alpha, \beta,\gamma\) satisfy the equation above, find the value of \(\sin^{2}\alpha + \sin^{2}\beta + \sin^{2}\gamma\).

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