Consider $e^{-i\frac{\pi}{2}}$. Using exponent rules, translating to $x+iy$ form and simplifying gives:
$$ e^{-i \frac{\pi}{2}} = e^{-1}e^{i \frac{\pi}{2}} = e^{-1} (0+i) = \frac{i}{e}$$
On the other hand,
$$ e^{-i \frac{\pi}{2}} = (0 - i) = -i $$
Therefore,
$$ \frac{i}{e} = -i \implies \frac{1}{e} = -1 $$
Where is the faulty step and why is it faulty?
It's wrong from the start. What makes you think that$$e^{-i\frac\pi2}=e^{-1}e^{i\frac\pi2}?$$