<prompt> Kokotoa tanleftfracpi7righttanleftfrac2pi7righttanleftfrac3pi 7kulia.  <response>: Kwa ujumla, Na nadharia ya DeMoivre,anzaalignjina la operetacis n theta jina la operetacis thetan cos theta i sin thetan cosn theta binomn1 i cosn - 1 theta sin theta - binomn2 cosn - 2 theta sin2 theta - binomn3 i cosn - 3 theta sin3 theta dotsb.endalignKulingana na sehemu halisi na za kufikirika, tunapataanzaaligncos n theta cosn theta - binomn2 cosn - 2 theta sin2 theta binomn4 cos n - 4 theta sin4 theta - dotsb, sin n theta binomn1 cosn - 1 theta sin theta - binomn3 cosn - 3 theta sin 3 theta binomn5 cosn - 5 theta sin5 theta - dotsb.mwishoalignKwa hiyo,anzaaligntan n theta fracsin n thetacos n theta fracdbinomn1 cosn - 1 theta sin theta - dbinomn3 cosn - 3 theta sin3 theta dbinomn5 cosn - 5 theta sin5 theta - dotsbcosn theta - dbinomn2 cos n - 2 theta sin2 theta dbinomn4 cosn - 4 theta sin4 theta - dotsb fracdbinomn1 tan theta - dbinomn3 tan3 theta dbinomn5 tan5 theta - dotsb 1 - dbinomn2 tan2 theta dbinomn4 tan4 theta - dotsb.endalignTunachukua n 7, tunapatatan 7 theta frac7 tan theta - 35 tan3 theta 21 tan5 theta - tan7 theta1 - 21 tan2 theta 35 tan4 theta - 7 tan6 theta.Kumbuka kwamba kwa theta fracpi7, frac2 pi7, na frac3 pi7, tan 7 theta 0. Hivyo, tan fracpi7, tan frac2 pi 7, na tan frac3 pi7 ndio mizizi ya7t - 35t3 21t5 - t7 0,au t7 - 21t5 35t3 - 7t 0. Tunaweza kuchukua kipengele cha t, kupatat t6 - 21t4 35t2 - 7 0.Tunajua kwamba mizizi mitatu ni tan fracpi7, tan frac 2 pi7, na tan frac3 pi7. Kwa kuwa vielelezo katika t6 - 21t4 35t2 - 7 vyote ni sawa, mizizi mingine mitatu ni -tan fracpi7, -tan frac2 pi7, na -tan frac3 pi7 . Kisha kwa fomula za Vieta,kushoto tan fracpi7 kulia kushoto tan frac2 pi7 kulia kushoto tan frac3 pi 7 kulia kushoto -tan fracpi7 kulia kushoto -tan frac2 pi7 kulia kushoto -tan frac 3 pi7 kulia -7,hivyo tan2 fracpi7 tan2 frac2 pi7 tan2 frac3 pi7 7.Kwa kuwa wote pembe ni papo hapo, kila tangent ni chanya. Kwa hivyo,tan fracpi7 tan frac2 pi7 tan frac3 pi7 sqrt7. 