1.

If `sqrt((1+i)/(1-i))=(a+ib)` then show that `(a^(2)+b^(2))=1`.

Answer» We have
`(a+ib)=sqrt((1+i)/(1-i))=(sqrt(1+i))/(sqrt(1-i))xx(sqrt(1+i))/(sqrt(1+i))=((1+i))/(sqrt(1-i^(2)))`
`=((1+i))/(sqrt(2))=((1)/(sqrt(2))+(1)/(sqrt(2))i)`
`rArr" "|a+ib|^(2)=((1)/(sqrt(2)))^(2)+((1)/(sqrt(2)))^(2)=((1)/(2)+(1)/(2))=1`.
`rArr" "(a^(2)+b^(2)) = 1`.
Hence, `(a^(2)+b^(2))=1`.


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