The principal value $tan^{-1}\begin{Bmatrix}sin\left(-\frac{\pi}{2}\right)\end{Bmatrix},$ is

$\frac{\pi}{4}$ $\frac{-\pi}{4}$ $\frac{3\pi}{4}$ $-\frac{3\pi}{4}$

$\frac{-\pi}{4}$

We know that $sin\left(-\frac{\pi}{2}\right) = -1.$ $tan^{-1}\begin{Bmatrix}sin\left(-\frac{\pi}{2}\right)\end{Bmatrix}=tan^{-1}(-1)=-\frac{\pi}{4}$