The smallest positive integer `n` for which `((1+i)/(1-i))^n=1` is

(a)`8 `

(b) `16`

` (c) `12 `

(d) None of these

Explanation

`because -1 le (1+x^2)/(2x) le 1`
`rArr |(1+x^2)/(2x)| le 1 rArr (1+x^2)/(2|x|) le 1 rArr (1+|x|^2)/(2|x|)-1 le 0 rArr ((|x|-1)^2)/(|x|) le 0`
`because |x|>0 therefore (|x|-1)^2=0 rArr |x|=1 rArr x=pm1`
`rArr x=1 (therefore x > 0)`
`therefore ((1+i)/(1-i))^n=2/pi sin^(-1)(1)=2/pi pi/2 = 1 rArr (((1+i)^2)/2)^n=1 rArr (i)^n=1 `
`therefore` n=4 (least positive value )