(i) `z = (1+i)/(1-i)**(1+i)/(1+i) = (1+i^2+2i)/(1+1) = i = 0+i`
In polar form,
`r(cos theta + isin theta) = 0+i`
`=> rcos theta = 0 and rsin theta = 1`
`=> r^2(cos^2 theta+sin^2theta) = 0^2+1^2`
`=> r^2 = 1=> r = 1`
Now, `cos theta = 1 => theta = pi/2`
so, modulus is `1` and argument is `pi/2`.
(ii)`z = 1/(1+i) = 1/(1+i)**(1-i)/(1-i) = (1-i)(1+1) = (1/2-1/2i)`
In polar form,
`r(cos theta + isin theta) = 1/2-1/2i`
`=> rcos theta = 1/2 and rsin theta = -1/2`
`=> r^2(cos^2 theta+sin^2theta) = (1/2)^2+(-1/2)^2`
`=> r^2 = 1/2=> r = 1/(sqrt2)`
Now, `sin theta =-1/sqrt2 => theta = -pi/4`
so, modulus is `1/sqrt2` and argument is `-pi/4`.
