<prompt> Kamadhambi x cos x tan x cot x sec x csc x 7,kisha tafuta sin 2x.  <response>: Kueleza kila kitu kulingana na sin x na cos x, tunapatasin x cos x fracsin xcos x fraccos xsin x frac1sin x frac 1cos x 7.Kishadhambi x cos x fracsin2 x cos2 xsin x cos x fracsin x cos xsin x cos x 7,ambayo inakuwasin x cos x fracsin x cos xsin x cos x 7 - frac1sin x cos x.Tunaweza weka upande wa kushoto, na ubadilishe sin x cos x na frac12 sin 2x:sin x cos x kushoto 1 frac2sin 2x kulia 7 - frac2sin 2x.Kwa hiyo,sin x cos xsin 2x 2 7 dhambi 2x - 2.Kuweka pande zote mbili, tunapatadhambi2 x 2 dhambi x cos cos2 xdhambi2 2x 4 dhambi 2x 4 49 dhambi2 x - 28 dhambi x 4 .Tunaweza kuandika hii kamadhambi 2x 1sin2 2x 4 dhambi 2x 4 49 sin2 x - 28 dhambi x 4.Hii hurahisishasin3 2x - 44 sin2 2x 36 dhambi 2x 0,hivyo dhambi 2x sin2 2x - 44 dhambi 2x 36 0.Ikiwa sin 2x 2 sin x cos x 0, basi usemi kwenye shida huwa haufafanuliwa. Vinginevyo,sin2 2x - 44 sin 2x 36 0.Kwa fomula ya quadratic,sin 2x 22 pm 8 sqrt7.Kwa kuwa 22 8 sqrt7 1, lazima tuwe na sin 2x 22 - 8 sqrt7. 