Tangents of sums

Geometry Level 3

Let \(e_k\) (for \(k = 0, 1, 2, 3, ...\)) be the \(k\)th-degree elementary symmetric polynomial in the variables

\[x_i= \tan \theta_i\]

for \(i = 0, 1, 2, \ldots\) i.e.

\[e_0 = 1\]

\[e_1 = \displaystyle \sum_i \tan \theta_i\]

\[e_2 =\displaystyle \sum_{i<j} \tan \theta_i \tan \theta_j\]

and so forth.

Find the value of

\[\tan \left( \displaystyle \sum_i \theta_i \right) .\]

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