<prompt> Acha A,B,C ziwe pembe za pembetatu. Tathminianzavmatrix sin2 A cot A 1 sin2 B cot B 1 sin2 C cot C 1 mwishovmatrix.  <response>: Tunaweza kupanua kibainishi kama ifuatavyo:anzaalignanzavmatrix sin2 A cot A 1 sin2 B kitanda B 1 dhambi2 C kitanda C 1 mwishovmatrix dhambi2 A anzavmatrix kitanda B 1 kitanda C 1 mwishovmatrix - cot A anzavmatrix sin2 B 1 sin2 C 1 mwishovmatrix anzavmatrix dhambi2 B kitanda B dhambi2 C kitanda C mwishovmatrix dhambi2 A cot B - cot C - cot A sin2 B - sin2 C sin2 B cot C - cot B sin 2 C sin2 A cot B - cot C sin2 B cot C - cot A sin2 C cot A - cot B.mwishoalignKwa ujumla,anzaaligncot x - cot y fraccos xsin x - fraccos ysin y fraccos x sin y - sin x cos ysin x sin y fracsin y - xsin x sin y.endalignKisha kibainishi ni sawa naanzaalignsin2 A cot B - cot C sin2 B cot C - cot A sin2 C cot A - cot B sin2 A cdot fracsin C - Bsin B sin C sin2 B cdot fracsin A - Cdhambi A dhambi C dhambi2 C cdot fracdhambi B - Asin A dhambi B fracsin3 A dhambi C - B sin3 B dhambi A - C sin3 C dhambi B - Adhambi A dhambi B dhambi C.mwishoalignSasa,anzaalignsin3 A dhambi A sin2 A dhambi 180circ - B - C sin2 A dhambi B C dhambi2 A,mwishoalignhivyo sin3 A dhambi C - B sin2 A dhambi C - B dhambi B C. Kishaanzaaligndhambi C - B dhambi B C dhambi C cos B - cos C dhambi Bdhambi B cos C cos B dhambi C cos B dhambi B cos C dhambi C cos2 B dhambi2 C - dhambi2 B cos2 C - cos B dhambi B cos C dhambi C cos2 B dhambi2 C - dhambi2 B cos2 C 1 - dhambi2 B dhambi2 C - dhambi2 B 1 - dhambi2 C dhambi2 C - dhambi2 B dhambi2 C - dhambi2 B dhambi2 B dhambi2 C dhambi2 C - dhambi2 B,mwishoalignhivyosin3 A dhambi C - B sin2 A sin2 C - sin2 B.Vile vile;anzaaligndhambi3 B dhambi A - C dhambi2 B dhambi2 A - dhambi2 C, dhambi3 C dhambi B - A dhambi2 C dhambi2 B - dhambi2 A.mwishoalignKwa hiyo,anzaaligndhambi3 A dhambi C - B dhambi3 B dhambi A - C dhambi3 C dhambi B - A dhambi2 A sin2 C - sin2 B dhambi2 B sin2 A - sin2 C dhambi2 C sin2 B - dhambi2 A 0,endalignambayo ina maana kibainishi ni sawa na 0. 