Writing $e^{i\theta}(e^{in\theta}-1)/(e^{i\theta}-1)$ in $(a+i b)$ form

Question

How to write:

$$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}$$ in

$$(a+i b) $$ $$ ?$$

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I tried to multiplicate by $$e^{i}$$ (the numerator and denominator). And to use $$i^2 = -1$$

But I got nothing.

Answer 1

Hint. We have $$e^{ni\theta}-1=e^{ni\theta/2}(e^{ni\theta/2}-e^{-ni\theta/2}) =e^{ni\theta/2}(2i\sin(n\theta/2))\ .$$ Do the same for the denominator, collect exponentials, and you get something of the form $e^{i\alpha}$ times a real number, which should then be easy.

Answer 2

Hint:

Write the denominator as $$e^{i\theta}-1 =(\cos(\theta)-1)+i\sin(\theta)$$

and then multiply both numerator and denominator by $$(\cos(\theta)-1)-i\sin(\theta)$$