If $a = \cos{ heta + i\sin heta}$, then $\frac{1 + a}{1 - a}$ is equal to

Question

If $a = \cos{	heta + i\sin	heta}$, then $\frac{1 + a}{1 - a}$ is equal to

Dr. Abubacker · Accepted Answer

(a)
Given, $a = \cos{	heta + i\sin	heta}$
Now, $\frac{1 + a}{1 - a} = \frac{1 + \cos{	heta + i\sin	heta}}{1 - \cos{	heta - i\sin	heta}}$
$= \frac{\left( 1 + \cos	heta ight) + i\sin	heta}{\left( 1 - \cos	heta ight) - i\sin	heta} 	imes \frac{\left( 1 - \cos	heta ight) + i\sin	heta}{\left( 1 - \cos	heta ight) + i\sin	heta}$
$= \frac{\sin^{2}{	heta + 2i\sin{	heta - \sin^{2}	heta}}}{1 + \cos^{2}{	heta - 2\cos{	heta + \sin^{2}	heta}}}$
$= \frac{i\ 4\sin{\frac{	heta}{2}.\cos\frac{	heta}{2}}}{4\sin^{2}\frac{	heta}{2}} = i\cot\frac{	heta}{2}$

Q. If $a = \cos{\theta + i\sin\theta}$, then $\frac{1 + a}{1 - a}$ is equal to

Explanation

Reviewed by Dr. Abubacker

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