<prompt> Kokotatan 5circ tan 25circ tan 45circ dots tan 165circ.  <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 9, tunapatatan 9 theta frac9 tan theta - 84 tan3 theta 126 tan5 theta - 36 tan7 theta tan9 theta1 - 36 tan2 theta 126 tan4 theta - 84 tan6 theta 9 tan8 theta.Kumbuka kwamba kwa theta 5circ, 25 circ, dots, 165circ, tan 9 theta tan 45circ 1. Hivyo,1 frac9 tan theta - 84 tan3 theta 126 tan5 theta - 36 tan7 theta tan9 theta1 - 36 tan 2 theta 126 tan4 theta - 84 tan6 theta 9 tan8 theta.Hebu t tan theta, hivyo1 frac9t - 84t3 126t5 - 36t7 t91 - 36t2 126t4 - 84t6 9t8.Hivyo, tan 5circ, tan 25circ, dots, tan 165circ ndio mizizi yat9 - 9t8 - 36t7 84t6 126t5 - 126t4 - 84t3 36t2 9t - 1 0.Kwa fomula za Vieta, jumla yao ni 9 . 