If the direction cosine of two lines are such that $l+m+n=0,l^2+m^2-n^2=0$the angle between them is ______

$l+m+n=0\dots ..\left(1\right)$

$l^2+m^2-n^2=0\dots .\left(2\right)$

$\Rightarrow n=-l-m$

$\Rightarrow l^2+m^2-{(-l-m)}^2=0$

$\Rightarrow -2lm=0$

$l=0$ and $m=0$
Case (1) : $l=0\Rightarrow m+n=0$
$\Rightarrow m=-n$
$\Rightarrow \frac{m}{-1}=\frac{n}{1}=\frac{l}{0}\Rightarrow (0,-1,1)$
Case (2) : $m=0\Rightarrow l+n=0$$l=-n\Rightarrow (-1,0,1)$

$\frac{l}{-1}=\frac{m}{0}=\frac{n}{1}$

$Cos\theta =\frac{1}{2}\Rightarrow \theta =\frac{\pi }{3}$