# 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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