<prompt> Kuna nambari halisi a na b hivi kwamba kwa kila nambari chanya x, tuna kitambulisho tan-1 kushoto frac1x - fracx8 kulia tan-1shoka tan-1 bx fracpi2 , .Katika mlingano huu wote, tan-1 ina maana ya kitendakazi cha tanjiti kinyume, ambacho wakati mwingine huandikwa arctan. Thamani ya a2 b2 ni ipi?  <response>: Kwa formula ya kuongeza tangent,tan tan-1 p tan-1 q fracp q1 - pq.Kutoka kwa mlinganyo uliotolewa,tan-1 shoka tan-1 bx fracpi2 - tan-1 kushoto frac1x - fracx8 kulia.Kishatan tan-1 shoka tan-1 bx tan kushoto fracpi2 - tan-1 kushoto frac1x - fracx8 kulia kulia,Upande wa kushoto ni fracax bx1 - abx2. Kulia - mkono ni upandeanzaaligntan kushoto fracpi2 - tan-1 kushoto frac1x - fracx8 kulia kulia frac1tan left tan-1 kushoto frac1x - fracx8 kulia kulia frac1frac1x - fracx8 fracx1 - frac18 x2.endalignKwa hivyo, a b 1 na ab frac18, hivyoa2 b2 a b2 - 2ab 1 - frac28 frac34. 