1+i1−i4+1−i1+i4==1+i1−i×1+i1+i4+1−i1+i×1−i1+i4=1+i2+2i12+124+1+i2−2i12+124=2i24+−2i24=i4+−i4=1+1=2

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${(\frac{1 + i}{1 - i})}^{4} + {(\frac{1 - i}{1 + i})}^{4} =$

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Detailed Solution

${(\frac{1 + i}{1 - i})}^{4} + {(\frac{1 - i}{1 + i})}^{4} =$

$= {(\frac{1 + i}{1 - i} \times \frac{1 + i}{1 + i})}^{4} + {(\frac{1 - i}{1 + i} \times \frac{1 - i}{1 + i})}^{4}$

$= {(\frac{1 + i^{2} + 2 i}{1^{2} + 1^{2}})}^{4} + {(\frac{1 + i^{2} - 2 i}{1^{2} + 1^{2}})}^{4}$

$= {(\frac{2 i}{2})}^{4} + {(\frac{- 2 i}{2})}^{4}$

$= i^{4} + {(- i)}^{4} = 1 + 1 = 2$

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1+i1−i4+1−i1+i4=

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${(\frac{1 + i}{1 - i})}^{4} + {(\frac{1 - i}{1 + i})}^{4} =$