How to show $i^{-1} = -i$? [duplicate]
By definition, $i^{-1}$ is the (unique) complex number $a$ such that $a\cdot i=i\cdot a=1$.
Since $(-i)\cdot i=i\cdot (-i)=1$, we have $i^{-1}=-i$.
You can do this: $$ \frac{1}{i}=\frac{1}{i}\cdot\frac{i}{i}=\frac{i}{i^2}=\frac{i}{-1}=-i. $$
$i^{-1}=i^{3-4}=\frac{i^3}{i^4}=i^3=-i$