Question

If z is a non-real complex number lying on the circle | z | = 1, then z is equal to

• A. $\frac{1-itan[\frac{argz}{2}]}{1+itan\left[\frac{argz}{2}\right]}$
• B. $\frac{1+itan[\frac{argz}{2}]}{1-itan[\frac{argz}{2}]}$
• C. $\frac{1-itan\left(argz\right)}{\left(argz\right)}$
• D. none of these

Answer 1

The correct option is B $\frac{1+itan[\frac{argz}{2}]}{1-itan[\frac{argz}{2}]}$

Since |z| = 1,
$\therefore$ let $z=\cos \alpha +i\sin \alpha$

$\therefore z=\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}+\frac{2i\tan \frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}=\frac{1-{\tan }^2\frac{\alpha }{2}}{1+{\tan }^2\frac{\alpha }{2}}$

$=\frac{{(1+i\tan \frac{\alpha }{2})}^2}{(1+i\tan \frac{\alpha }{2})(1-i\tan \frac{\alpha }{2})}$

$=\frac{1+i\tan \left(\frac{argz}{2}\right)}{1-i\tan \left(\frac{argz}{2}\right)}(\because argz=\alpha )$