If the direction cosines of a line are $k$, $k$ and $k$, then

Since, direction cosines of a line are $k$, $k$ and $k$.

$\therefore$ $l = k,m = k$ and $n = k$

As we know, ${l^2} + {m^2} + {n^2} = 1$

$\Rightarrow$ ${k^2} + {k^2} + {k^2} = 1$

$\Rightarrow$ ${k^2} = \frac{1}{3}$

$\therefore k = \pm \frac{1}{{\sqrt 3 }}$