A complex rational equation I'm having trouble with

I've posted before about my fractal project, and this is something else that is related. The function:

$\displaystyle f(z)=\frac{1-z}{(2-z)^{\frac{3}{2}}}$ where $\displaystyle |z|=1$.

The derivative along this line:

$\displaystyle f'(z)=-\frac{ie^{it}(e^{it}+1)\sqrt{2-e^{it}}}{2(2-e^{it})^{3}}$ for a new real variable $\displaystyle t$.

I'm trying to find when the real and/or imaginary parts equal zero.

$\displaystyle \Re\{f'\}=0$

$\displaystyle \Im\{f'\}=0$

Is there any other way to solve these equations without splitting $\displaystyle f'$ into $\displaystyle x'$ and $\displaystyle iy'$ just because it is so messy and long? I know one method of removing the imaginary terms from the bottom by multiplying top and bottom by $\displaystyle (\frac{1}{2}-e^{it})^3$, factoring $\displaystyle e^{3it}$ out of the denominator, and multiplying numerator and denominator by $\displaystyle \frac{1}{e^{3it}}$ resulting in a denominator of $\displaystyle 4\cos t-5$. It's still messy with a numerator of $\displaystyle -\frac{i}{e^{2it}}(e^{it}+1)(\frac{1}{2}-e^{it})^3\sqrt{2-e^{it}}$ but now, algebraically, the worst part is $\displaystyle \sqrt{2-e^{it}}$. I'd appreciate any suggestions.