<prompt> Kokotakushoto 1 cos frac pi8 kulia kushoto 1 cos frac 3 pi8 kulia kushoto 1 cos frac 5 pi8 kulia kushoto 1 cos frac 7 pi8 kulia.  <response>: Wacha tuite usemi tunajaribu kujumuisha x. Kisha tuna xleft 1 cos frac pi8 kulia kushoto 1 cos frac 3 pi8 kulia kushoto 1 cos frac 5 pi8 kulia kushoto 1 cos frac 7 pi8 kulia. Hii ni sawa na kushoto 1 cos frac pi8 kulia kushoto 1 cos frac 3 pi8 kulia kushoto 1 - cos frac pi8 kulia kushoto 1 - cos frac 3 pi8 kulia. Kwa tofauti ya miraba, hii ni sawa na left1-cos2 fracpi8kulialeft1-cos2 frac3pi8. kulia . Kwa kutumia cos2 asin2 a1, tuna left1-cos2 fracpi8kuliakushoto1-cos2 frac3pi8rightsin2 fracpi8sin2 frac3pi8. Sawa, sasa tunahitaji tu kuhesabu sines. Fomula hiyo ni sinasinb frac12leftcosa-b-cosabright. Hiyo ina maana kwamba sinfracpi8sinfrac3pi8 frac12leftcosfrac2pi8 -cosfrac4pi8kulia. Hii ni sawa na frac12leftcosfracpi4-cosfracpi2right. Hii ni sawa na frac12leftfracsqrt22-0right. Hii ni sawa na fracsqrt24. Kwa hivyo, leftsin2 fracpi8rightleftsin2 frac3pi8right leftfrac sqrt24kulia2 frac18. Kwa hivyo, xfrac18. 