Proving $\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$

Question

Prove that

$$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}=\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$$

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I tried to use $$ e^{in\theta} =\cos n\theta+i\sin n\theta$$ but with this way I got a big expression with cos(n*x)*sin^2(x)..... So. I'm lost.

Answer 1

This is trivially false: simply take $n = 1$, $\theta = \pi/2$.