I'm reading a book about the Discrete Fourier Transform. In one of the chapters (available here) discussing Spectral Leakage, I have difficulties following a derivation step. Specifically, I don't understand how:

$\displaystyle \frac{1-e^{j(\omega_x - \omega_k)NT}}{1-e^{j(\omega_x - \omega_k)T}} = e^{j(\omega_x-\omega_k)(N-1)T/2}\frac{sin[(\omega_x-\omega_k)NT/2]}{sin[(\omega_x-\omega_k)T/2]}$

where $\displaystyle j$ is the imaginary unit $\displaystyle \sqrt{-1}$,

$\displaystyle T$ is a positive real value (sampling period),

$\displaystyle N$ is a positive integer (the number of samples),

$\displaystyle \omega_k = 2\pi\frac{k}{N}f_s$ for $\displaystyle k \in [0, N-1]$ and $\displaystyle f_s = \frac{1}{T}$ is the sampling frequency,

$\displaystyle \omega_x$ is a positive real value (AFAIK, is an angle in radians but is not any particular $\displaystyle \omega_k$, as defined above -- please correct me if I'm wrong)

My algebra is a bit weak so a hint is appreciated!