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In what should come as no surprise, you don't determine the number of normal modes by the number of $\cos$ subexpressions occurring in the solution. (I guess there's a sense where you could say that you're "counting the cosines", but it's not this simple syntactic sense.)

It would help to have the Big Picture behind normal modes and why we care about them. Let $L$ be a constant coefficient, linear differential operator. For example, the early examples in the chapter you link would have $L = m\frac{d^2}{dt^2} + K$ where $K = \begin{pmatrix} k + \kappa & -\kappa \\ -\kappa & k + \kappa\end{pmatrix}$. Solving the differential equation then means finding a function $f$ satisfying the boundary conditions such that $Lf=0$.

Write $\mathcal F[f](\omega) = \int_{-\infty}^\infty f(t)e^{-i\omega t}dt$ for the [Fourier transform](https://en.wikipedia.org/wiki/Fourier_transform) of $f$. Using the key property $\mathcal F\left[\frac{df}{dt}\right](\omega) = i\omega\mathcal F[f](\omega)$, our differential equation $Lf = 0$ becomes $$Lf = 0 \iff \mathcal F[Lf](\omega) = 0 \iff \tilde L(i\omega)\mathcal F[f](\omega) = 0$$
where $\tilde L(i\omega)$ is a matrix whose components are polynomials in $i\omega$. For example, the above $L$ becomes $\tilde L(i\omega) = \begin{pmatrix}m(i\omega)^2 + k + \kappa & -\kappa \\ -\kappa & m(i\omega)^2 + k + \kappa\end{pmatrix}$ which is equation $(9)$ in the paper. $\tilde L(i\omega)$ for a fixed $\omega$ is a linear operator, and, indeed, $\mathcal F$ is a linear operator. We get that a normal mode corresponds to an orthogonal basis vector of the linear subspace $\tilde L(i\omega)^{-1}(\mathbf 0)$ for some $\omega$. Altogether, they form an orthogonal basis for the linear subspace of solutions to $Lf = 0$. Indeed, they essentially let us build a solution in terms of its Fourier components.

From equation $(10)$, we see if $\kappa=0$, there is only one frequency $\omega$ (modulo sign), and $A_1$ and $A_2$ can be chosen independently. This makes sense since the equations then describe two independent oscillators. This would lead to only one "cosine", but we'd have two distinct normal modes at the single frequency.

For the equation you list at the end, the reason there are infinitely many normal modes is *any* frequency $\omega$ can be chosen which will lead to $k = \omega\sqrt{\rho/E}$ as written in equation $(84)$. So there's a distinct solution for any frequency. The logic used in the section can again be analyzed in terms of Fourier transforms. In this case, we'd use a two-dimensional Fourier transform satisfying $\mathcal F\left[\frac{\partial f}{\partial x}\right](\omega,k) = ik\mathcal F[f](\omega,k)$ and $\mathcal F\left[\frac{\partial f}{\partial t}\right](\omega,k) = i\omega\mathcal F[f](\omega,k)$ which would transform $E\frac{\partial^2 f}{\partial x^2} - \rho\frac{\partial^2 f}{\partial t^2} = 0$ to the equation $-E k^2 + \rho\omega^2 = 0$ which is just equation $(84)$ again.

If you're not familiar with Fourier transforms, then the above is probably hard to go through. In that case, though, I'll quote from the final paragraph of the chapter you linked:
> Fourier analysis plays an absolutely critical role in the study of waves. In fact, it is so important that we’ll spend all of Chapter 3 on it.

Fourier analysis is an extremely valuable tool across all of math and physics, both theoretical and practical. It should be clear from the above that it simplifies and systematizes the computation of normal modes, or, indeed, the solution to this class of differential equations in general.