If a line has direction ratios $⟨a+b,b+c,c+a⟩$, then what is the sum of the squares of its direction cosines?

Direction cosines $(l,m,n)$ of any line always satisfy the fundamental identity:
$$l^2+m^2+n^2=1$$
This is true regardless of what the direction ratios are. Direction ratios are proportional to direction cosines, and the normalization process $l=\frac{a}{\sqrt{a^2+b^2+c^2}}$ ensures that the sum of their squares is always unity.