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

Explanation:

Step1: Recall the property of inverse - tangent function

The inverse - tangent function $y = \tan^{-1}(x)$ has a range of $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$. Also, $\tan^{-1}(\tan(x))=x$ when $x\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$.

Step2: Check the value of $x =-\frac{\pi}{4}$

The value $x =-\frac{\pi}{4}$ lies in the interval $\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ since $-\frac{\pi}{2}\approx - 1.57$ and $\frac{\pi}{2}\approx1.57$ and $-\frac{\pi}{4}=- 0.785$.

Step3: Apply the property

Since $-\frac{\pi}{4}\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$, we have $\tan^{-1}\left(\tan\left(-\frac{\pi}{4}\right)\right)=-\frac{\pi}{4}$.

Answer:

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