<prompt> Tuseme kazi psi inatosheleza psi1 sqrt2 sqrt2 sqrt2 napsi3x 3 psix psi3xkwa x. zote halisi Amua prod_n 1100 psi3n .  <response>: Tunaweza kuandika sqrt2 2 cos fracpi4. Kwa fomula ya nusu-pembe,sqrt2 sqrt2 sqrt2 2 cos fracpi4 2 cos fracpi8,napsi1 sqrt2 sqrt2 sqrt2 sqrt2 2 cos fracpi8 2 cos frac pi16.Sasa, tuseme psix 2 cos theta kwa pembe fulani theta. Kishaanzaalignpsi3x psi3x - 3 psix 8 cos3 theta - 6 cos theta 2 cos 3 theta.endalignKwa kuwa psi1 2 cos fracpi16, inafuata kwambapsi3n 2 cos frac3n cdot pi16kwa nambari zote chanya n. Kishaanzaalignpsi3 2 cos frac3 pi16, psi32 2 cos frac9 pi16, psi33 2 cos frac27 pi16 -2 cos frac11 pi16, psi34 2 cos frac81 pi16 -2 cos fracpi16, psi35 2 cos frac243 pi16 -2 cos frac3 pi16, psi36 2 cos frac729 pi16 -2 cos frac9 pi16, psi37 2 cos frac2187 pi16 2 cos frac11 pi16, psi38 2 cos frac6561 pi16 2 cos fracpi16.mwishoalignKwa hiyo,anzaalignpsi3 psi32 psi33 psi34 kushoto 2 cos frac3 pi16 kulia kushoto 2 cos frac9 pi16 kulia kushoto 2 cos frac11 pi16 kulia kushoto 2 cos fracpi16 haki kushoto 2 cos frac3 pi16 kulia kushoto -2 sin fracpi16 kulia kushoto -2 sin frac 3 pi16 kulia kushoto 2 cos fracpi16 kulia 4 cdot 2 sin fracpi16 cos fracpi16 cdot 2 sin frac3 pi16 cos frac3 pi 16 4 dhambi fracpi8 dhambi frac3 pi8 4 dhambi fracpi8 cos fracpi8 2 sin fracpi4 sqrt2.mwishoalignVile vile, psi35 psi36 psi37 psi38 sqrt2. Zaidi ya hayo, psi 34 -psi1, hivyo psi3n psi3n 1 psi3n 2 psi3 n 3 sqrt2 kwa nambari zote chanya n. Kwa hivyo,prod_n 1100 psi3n sqrt225 4096 sqrt2. 