The domain of definition of $f\left(x\right)={\log }_x\cos x$, is $(0,\pi /2)-\left\{k\right\}$ then k is 

For $f\left(x\right)$ to be real, we must have

$\cos x>0,x>0$ and $x\ne 1$

$\Rightarrow -\pi /2<x<\pi /2,x>0$ and $\mid x\ne 1\Rightarrow x\in (0,\pi /2)-\left\{1\right\}$

Hence, domain $\left(f\right)=(0,\pi /2)-\left\{1\right\}$