Question on complex number calculation for transmission coefficient of finite potential well

Question

This is actually in my quantum mechanics textbook (pure math question though), and I just cannot see why this equality is true. Any help would be greatly appreciated!

Let $F$ and $A$ be nonzero complex numbers, and let $a, k, \text{ and}\ l$ be positive real numbers. Assume that

\begin{equation} F = \frac{e^{-2ika} A}{\cos(2la) - i \frac{k^2 + l^2}{2kl} \sin(2la)}. \end{equation}

Define $T := |F|^2 / |A|^2.$ Then

\begin{equation} T^{-1} = 1 + \left(\frac{k^2 - l^2}{2kl}\right)^2 \sin^2(2la). \end{equation}

Thanks so much!

Answer 1

Hint:

$$\left ( \frac{k^2+\ell^2}{2 k \ell} \right )^2 = 1 + \left ( \frac{k^2-\ell^2}{2 k \ell} \right )^2$$