find the exact value of the expression, if it is defined. (if an answer is undefined, enter undefined.)…

Explanation:

Step1: Let $\theta=\tan^{-1}(-1)$

By the definition of the inverse - tangent function, $\tan\theta=-1$ and $\theta\in(-\frac{\pi}{2},\frac{\pi}{2})$. The angle $\theta$ in the interval $(-\frac{\pi}{2},\frac{\pi}{2})$ for which $\tan\theta = - 1$ is $\theta=-\frac{\pi}{4}$.

Step2: Find $\sin(\tan^{-1}(-1))$

Since $\tan^{-1}(-1)=-\frac{\pi}{4}$, then $\sin(\tan^{-1}(-1))=\sin(-\frac{\pi}{4})$. We know that $\sin(-x)=-\sin x$, so $\sin(-\frac{\pi}{4})=-\sin\frac{\pi}{4}$. And $\sin\frac{\pi}{4}=\frac{\sqrt{2}}{2}$, so $\sin(-\frac{\pi}{4})=-\frac{\sqrt{2}}{2}$.

Answer:

$-\frac{\sqrt{2}}{2}$