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

$\tan \left( \displaystyle \sum_i \theta_i \right) .$

\dfrac {e_1 - e_3 + e_5 - \ldots}{e_0 - e_2 + e_4 - \ldots}

\dfrac {e_0 - e_2 + e_4 - \ldots}{e_1 + e_3 - e_5 + \ldots}

\dfrac {e_0 - e_2 + e_4 - \ldots}{e_1 - e_3 + e_5 - \ldots}

\dfrac {e_1 + e_3 - e_5 + \ldots}{e_0 - e_2 + e_4 - \ldots}