Tangents of sums

Geometry Level 3

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

xi=tanθix_i= \tan \theta_i

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

e0=1e_0 = 1

e1=itanθie_1 = \displaystyle \sum_i \tan \theta_i

e2=i<jtanθitanθje_2 =\displaystyle \sum_{i<j} \tan \theta_i \tan \theta_j

and so forth.

Find the value of

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

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