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Q.
Let $\alpha$ be the angle between the lines whose direction cosines satisfy the equations
$l+m-n=0$ and $l^{2}+m^{2}-n^{2}=0 .$ Then the value of $\sin ^{4} \alpha+\cos ^{4} \alpha$ is
$n =\ell+ m$
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, $\cos \alpha=\frac{1}{2} \Rightarrow \alpha=\frac{\pi}{3}$