The direction cosines  $\mathcal{l},m$  and $n$ of two lines are connected by the relations  $\mathcal{l}+m+n=0$  and  $\mathcal{l}m=0$ , then the angle between the lines is 

Given $\mathcal{l}+m+n=0$ …(i)
and  $\mathcal{l}m=0$
i.e.,either m=0 or  $\mathcal{l}=0$
If  $\mathcal{l}=0$ , then putting in equation (i) we get m=-n
$\therefore$  Direction ratios are 0,-n,n i.e.0,-1,1
If m=0, then putting in equation(i),we get $\mathcal{l}=-n$
$\therefore$  Direction ratios are –n,0,n i.e.-1,0,1
 $\cos \theta =\frac{0\times (-1)+(-1)\times 0+1\times 1}{\sqrt{{\left(0\right)}^2+{(-1)}^2+{\left(1\right)}^2}\sqrt{{(-1)}^2+{\left(0\right)}^2+{\left(1\right)}^2}}=\frac{1}{2}$
 
$\Rightarrow \theta =\frac{\pi }{3}$