if (l1,m1,n1),(l2,m2,n2),(l3,m3,n3) are the direction cosines of three mutual perpendicular lines,show that the line whose direction ratios area l1+l2+l3,m1+m2+m3,n1+n2+n3 make equal angles with them

Because the first three lines are mutually perpendicular, dot products of pairs of lines are zero. 
l1*l2+m1*m2+n1*n2 = 0 
l2*l3+m2*m3+n2*n3 = 0 
l1*l3+m1*m3+n1*n3 = 0 

Finally, using the above equations, and another dot product, it is shown that the cosine of the angle between the fourth vector and any of the first three vectors is the same.

Hence, it makes equal angles with them.