<prompt> Acha z_1, z_2, dots, z_20 iwe mizizi ishirini changamano ya mlinganyo.z20 - 4z19 9z18 - 16z17 dots 441 0.Hesabu cot left sum_k 1 20 operatornamearccot z_k kulia. Kumbuka kwamba fomula ya nyongeza ya kotangent bado ni halali inapofanya kazi na nambari changamano.  <response>: Tunaanza na formula ya kuongeza kwa tangent:tan a b fractan a tan b1 - tan a tan b.Kishaanzaalign kitanda a b frac1tan a b frac1 - tan a tan btan a tan b fracfrac1tan a tan b - 1frac1tan a frac1tan b fraccot a cot b - 1cot a cot b.mwishoalignKishaanzaalign kitanda a b c kitanda a b c fraccot a b cot c - 1cot a b cot c fracfraccot a cot b - 1cot a cot b cdot cot c - 1fraccot a cot b - 1cot a kitanda b kitanda c fraccot a cot b cot c - cot a cot b cot ccot a cot b cot a cot c cot b cot c - 1.endalignKwa ujumla zaidi, tunaweza kuthibitisha hilo kitanda a_1 a_2 vitone a_n fracs_n - s_n - 2 dotsbs_n - 1 - s_n - 3 dotsb, ambapo s_k ni jumla ya bidhaa za cot a_i, zinazochukuliwa k kwa wakati mmoja. Katika nambari, masharti ni s_n, s_n - 2, s_n - 4, s_n - 6, dots, na ishara mbadala. nambari inaishia s_0 1 au s_1, kulingana na kama n ni sawa au isiyo ya kawaida. Masharti katika kipunguzo yanafafanuliwa vile vile.Hebu a_i jina la operetaarccot z_i. Kisha kitanda a_1 a_2 vitone a_20 fracs_20 - s_18 vitone - s_2 1s_19 - s_17 nukta s_3 - s_1.Kwa fomula za Vieta, s_1 22, s_2 32, s_3 42, dots, s_19 20 2, na s_20 212. Kwa hiyo,anzaaligncot a_1 a_2 dots a_20 fracs_20 - s_18 dots - s_2 1s_19 - s_17 dots s_3 - s_1 frac212 - 192 172 - 152 vitone 52 - 32 1202 - 182 162 - 142 vitone 42 - 22 frac21 - 1921 19 17 - 1517 15 vitone 5 - 35 3 120 - 18 20 18 16 - 14 16 14 vidoti 4 - 2 4 2 frac221 19 17 15 dots 5 3 1220 18 16 14 dots 4 2 frac241220.mwishopanga 