Let z be a non-real complex number
lying on `|z|=1,` prove that `z=(1+itan((arg(z))/2))/(1-itan((arg(z))/(2)))` (where `i=sqrt(-1).)`

Options

A. `(1-itan((argz)/2))/(1+itan((argz)/2))`
B. `(1+itan((argz)/2))/(1-itan((argz)/(2)))`
C. `(1-itan(argz))/(1+itan((argz)/2))`
D. none of these

Answer

B

Explanation

We have,
`|z|=1`
`rArr z=costheta+isintheta`,
`rArr z=(1-tan^(2)theta/2)/(1+tan^(2)theta/2) +(2itantheta/2)/(1+tan^(2)theta/2)`
`z = (1+itantheta/2)^(2)/((1+itantheta/2)(1-itantheta/2))`
`rArr z=(1+itantheta/2)/(1-itantheta/2) rArr z=(1+itan(argz)/2)/(1-itan(argz)/(2))`