Consider Euler's Formula: $$e^{i\theta}=cos(\theta)+isin(\theta)$$ Using this, the value of $i$ is $e^{{i\pi}/2}$. Following this: $$i^i=e^{i^2\pi/2}$$ The accepted principal result of this is that $$i^i=e^{-\pi/2}$$ I appreciate that the result given is one of many solutions - but for my question here, I'm only focussing on this case.

Instead of simply using $i^2=-1$, I will separate the exponent for $i^i=e^{i*i\pi/2}$.

Going back to Euler's Formula, and inserting ${\theta=i\pi/2}$ we get: $$i^i=\cos (i\pi/2) + i\sin (i\pi/2)$$ Which gives $$cosh(\pi/2)+i*sinh(pi/2)$$ Or roughly $2.509+2.301i$ Again - I know there are many different solutions here, I'm just focussing on the principal case.

Why is there a contradiction here? What exactly causes my method to not yield the same result? Originally my answer was marked as duplicate - however the thread I was linked to did not contain this method - and the reasons for its failure was said to be due to $ln(i)$ not being well defined over complex space (I used $ln(i)$ in my original question) . Considering that I can now show my method without the use of $ln(i)$, I am wondering if there is another reason why my method is incorrect.