The angle between the lines whose direction cosines are given by   $1+\mathrm{m}+\mathrm{n}=0 \text{and }2{\mathrm{l}}^2+2\mathrm{m}+{\mathrm{n}}^2=0$

$1^2+m^2+n^2=1; 1+m+n=0 ; {21}^2+2m^2-n^2=0$  Solving above equations, we get the
direction cosines as $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},\sqrt{\frac{2}{3}}\right)$  and  $\left(\frac{1}{\sqrt{6}},\frac{1}{\sqrt{6}},-\sqrt{\frac{2}{3}}\right)$ or  $\left(-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}},\sqrt{\frac{2}{3}}\right)$ and $\left(-\frac{1}{\sqrt{6}},-\frac{1}{\sqrt{6}},-\sqrt{\frac{2}{3}}\right)$  The angle between these lines in both the cases is  ${\cos }^{-1}(-\frac{1}{3})$