The principal argument of

Let $z=i^{-1097}$

$z=\frac{1}{(i)^{1096} \cdot i}$

$=\frac{1}{\left(i^{4}\right)^{274}} \times \frac{1}{i}$

$=\frac{1}{1} \times \frac{1}{i} \quad\left(\because i^{4}=1\right)$

$=\frac{1}{i}$

$=\frac{1}{i} \times \frac{i}{i}$

$z=-i=\cos \left(\frac{\pi}{2}\right)-i \sin \frac{\pi}{2}$

Hence, principle argument is $-\frac{\pi}{2}$.