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Let eke_kek (for k=0,1,2,3,...k = 0, 1, 2, 3, ...k=0,1,2,3,...) be the kkkth-degree elementary symmetric polynomial in the variables
xi=tanθix_i= \tan \theta_ixi=tanθi
for i=0,1,2,…i = 0, 1, 2, \ldotsi=0,1,2,… i.e.
e0=1e_0 = 1e0=1
e1=∑itanθie_1 = \displaystyle \sum_i \tan \theta_ie1=i∑tanθi
e2=∑i<jtanθitanθje_2 =\displaystyle \sum_{i<j} \tan \theta_i \tan \theta_je2=i<j∑tanθitanθj
and so forth.
Find the value of
tan(∑iθi).\tan \left( \displaystyle \sum_i \theta_i \right) .tan(i∑θi).
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