Intuitive understanding of the suspension bridge

It is possible to determine the shape of the cable of a suspension bridge, where it is assumed that the total weight of the load, the cable and the roadway per horizontal distance is constant. The shape the bridge takes is parabolic and can be derived as demonstrated in here or here.

Let's assume a small section of the bridge as shown in the image below. The tension in the cable is denoted by $$T$$ and the load per horizontal distance is $$\rho$$ (the load of a tiny element $$dx$$ is $$w=\rho dx$$). In the links provided it was shown that the tension in the cable follows these relationships:

$$\frac{d(Tcos\theta)}{dx}=0,$$ $$\frac{d(Tsin\theta)}{dx}=\rho.$$

It appears that the horizontal component of the tension always remain constant $$Tcos\theta=const.$$, while the vertical component increases linearly as we move away from the vertex of the parabola (the vertex of the parabola coincides with the origin of our x-y axes).

This simple relationship suggests that there possibly is a way to understand the variation of the tension in the cable intuitively.

Say I wanted to derive the shape of the cable. One way to do it would be to analyse all the forces of an infinitesimal element $$dx$$, similarly to how it was done in the provided links. But had we known the fact that the horizontal component of the tension is constant, we could have derived the result almost instantly.

So my question is, without going through the mathematical derivations, why is the horizontal component of the tension constant?