The direction cosines of three mutually perpendicular straight lines are $〈l_1,m_1,n_1〉,〈l_2,m_2,n_2〉,〈l_3,m_3,n_3〉$. Then the direction cosines of a line which is equally inclined to the given three lines, are

Given  $〈l_1,m_1,n_1〉,〈l_2,m_2,n_2〉,〈l_3,m_3,n_3〉$are the direction cosines of three mutually perpendicular lines 

It implies that $\sum l_1{l}_2=\sum l_2{l}_3=\sum l_1{l}_3=0,\sum {l_1}^2=\sum {l_2}^2=\sum {l_3}^2=1$

Hence the direction cosines of line which is equally inclined to the above lines are $\left(\frac{l_1+l_2+l_3}{3},\frac{m_1+m_2+m_3}{3},\frac{n_1+n_2+n_3}{3},\right)$