यदि `(a+ib)=sqrt((1+i)/(1-i))`, तब सिद्ध कीजिए कि `a^(2)+b^(2)=1`

Explanation

`because" "a+ib=sqrt((1+i)/(1-i))=sqrt(((1+i)(1+i))/((1-i)(1+i)))=(1+i)/(sqrt(1^(2)-i^(2)))=(1+i)/(sqrt(2))` . . . (i)
एवं `(bar(a+ib))=(1)/(sqrt(2))(bar(1+i))rArra-ib=(1)/(sqrt(2))(1-i)` . . .(ii)
समीकरण (i) और (ii) का प्रयोग करने पर,
`(a+ib)(a-ib)=(1)/(2)(1+i)(1-i)=(1)/(2)(1^(2)-i^(2))=(1)/(2)(1+1)=((1)/(2)xx2)=1`
अतः, `a^(2)+b^(2)=1`