The angle between the lines whose direction cosines satisfy the equations l+m+n=0 and l²=m²+n² is π/6, π/4, π/3, π/2

Given l²=m²+n² and l+m+n=0
We know that l²+m²+n²=1
So we get l²+l²=1 ⇒l²=1/2 ⇒l=±1/√2
Let us consider l=1/√2
Since l+m+n=0, we get m+n=-l
By substituting l=-(m+n) in l²=m²+n², we get mn=0
Put n=0, then we have m=-l=∓1/√2
put m=0, then we have n=-l=∓1/√2
So two possible directional cosines are (1/√2,∓1/√2,0) and (1/√2,0,∓1/√2)
Therefore angle between them is cos⁻¹(1/2+0+0) = cos⁻¹(1/2) = π/3