The most general value of $\theta$ which satisfies both the equations $\tan \theta =-1$ and  $\cos \theta =1/\sqrt{2}$ is

$$\begin{gathered} \tan \theta is negative and \cos \theta is positive ,hence \theta lies in fourth quadrant. \\ \tan \theta =-1\Rightarrow \theta =\frac{7\mathrm{\pi }}{4} \end{gathered}$$
$\cos \theta =1/\sqrt{2}\Rightarrow \theta =\frac{7\pi }{4}$
Common solution set $=2n\pi +\frac{7\pi }{4}$