I have a question about the following statement from an article here. He states on page 8 that:

https://sci-hub.ru/10.1002/mana.201200319

Let $\alpha = \dfrac{\rho_{1}\lambda^{2}}{\kappa_{1} + i\kappa_{2}\lambda}$ where $\rho_{1}, \kappa_{1}, \kappa_{2} > 0$. then $$ \text{Cot}(i\alpha L) \to 1, \ \ \text{when} \ \ n \to \infty $$ where $L > 0$.

Well, firstly, I think he's changing $\alpha$ to $\alpha_{n}$ and leaving $n$ hidden. And doing some mathematical manipulations, I'm getting to

Let $x_{n} = \Re(\alpha_{n}L)$ and $y_{n} = \Im(\alpha_{n}L)$. Using some properties, I can get $$ \text{Cot}(i\alpha_{n} L) = \dfrac{\cos(x_n)\sin(x_n)}{\cosh^{2}(y_n) - \cos^{2}(x_n)} - \dfrac{\tanh(y_n)} {\frac{\cos^{2}(x_n)}{\cosh^{2}(y_n)} - 1} $$

From the equality above can I conclude what I want? Why?