Zenith angle dependence of muon flux from cosmic rays

I’ve been trying to understand this paper about zenith angle dependence of muon flux. In particular I want to understand where the equation $I(theta)approx I(0) cos^2theta$ comes from. The paper states that

The degree of attenuation
through ionization is directly proportional to the
length of the path that they travel; therefore the flux
will be inversely proportional to the path length.

and this is the place cosine comes from. Why is "attenuation" proportional to the path travelled? Shouldn’t it be exponential, since travelling through 1 km of atmosphere should lower the intensity by the same fraction no matter how far along the path that 1 km is?

If that simplified line of reasoning in the paper is not correct, is there a better theoretical explanation for the $cos^2 theta$ dependency? I’ve tried to read the referenced papers, but they only have a really complicated calculation using the energy spectrum of primary cosmic rays. Is there a way to derive this relation without using that?