Answer
$ 135^{\circ}$ and $315^{\circ}$.
Work Step by Step
$\cot\theta=-1$
$\tan\theta=\frac{1}{\cot\theta}=-1$
The value of $\tan \theta$ is negative, so $\theta$ may lie in either quadrant II or IV.
We know that $\tan 45^{\circ}=1$.
Recall: $-\tan x= \tan(180^{\circ}-x)$ and $-\tan x= \tan(360^{\circ}-x)$
$\implies -\tan 45^{\circ}=-1=\tan(180^{\circ}-45^{\circ})$
$=\tan 135^{\circ}$
$135^{\circ}$ which is in the second quadrant is one value of $\theta$.
Now $-\tan 45^{\circ}=-1= \tan (360^{\circ}-45^{\circ})$
$=\tan 315^{\circ}$
$315^{\circ}$ is in the fourth quadrant.
The values of $\theta$ are $ 135^{\circ}$ and $315^{\circ}$.
