Now, $\ell^{2}+ m ^{2}= n ^{2}=(\ell+ m )^{2}$
$\Rightarrow 2 \ell m =0$
If $\ell=0 \Rightarrow m = n =\pm \frac{1}{\sqrt{2}}$
And, If $m=0 \Rightarrow n=\ell=\pm \frac{1}{\sqrt{2}}$
So, direction cosines of two lines are
$\left(0, \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}\right)$ and $\left(\frac{1}{\sqrt{2}}, 0, \frac{1}{\sqrt{2}}\right)$
Thus, $\cos \alpha=\frac{1}{2} \Rightarrow \alpha=\frac{\pi}{3}$
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