The acute angle between two lines whose direction cosines are given by the relation between $l+m+n=0$ and $l^2+m^2-n^2=0$  is

l+m+n=0
$l^2+m^2-n^2=0$ 
From Eqs. (i) and (ii),
 $l^2+m^2-{(-l-m)}^2=0$  
   lm = 0
l = 0 or m = 0
Case I : If, l=0
Then,    m+n=0
$\therefore \frac{l}{0}=\frac{m}{-1}=\frac{n}{1}=\frac{1}{\sqrt{2}}$  
Case II : If, m = 0
Then, l + n = 0
$\frac{l}{-1}=\frac{m}{0}=\frac{n}{1}=\frac{1}{\sqrt{2}}$ 
$\cos \theta =0+0+\frac{1}{2}$ 
 $\theta =\pi /3$