Answer
$135^{\circ}$ and $225^{\circ}$
Work Step by Step
$\cos\theta= \frac{1}{\sec\theta}=-\frac{1}{\sqrt 2}$
$\cos\theta$ is negative in the second and third quadrants.
$\cos 45^{\circ}=\frac{1}{\sqrt 2}$
Recall that $\cos (180^{\circ}-x)=\cos(180^{\circ}+x)=-\cos x$
$\implies -\cos45^{\circ}=-\frac{1}{\sqrt 2}=\cos(180^{\circ}-45^{\circ})=\cos 135^{\circ}$ and
$-\frac{1}{\sqrt 2}=\cos(180^{\circ}+45^{\circ})=\cos 225^{\circ}$
Values of $\theta$ are $135^{\circ}$ and $225^{\circ}$
