I have the function $f(n)= \pi (n+\frac{1}{2})+\epsilon \frac{2\pi(n+1/2)-a}{2\pi(n+1/2)}$ when $\epsilon<<1$, I can't understand what identity I need to use to prove that:
$\tan(f(n))\approx \frac{2\pi(n+1/2)}{\epsilon(2\pi(n+1/2)-a)}$ I tried with taylor but the first element in $f(x)$ taking tan function to $\infty$
The first step is to use
$$ \tan(x +n\pi +\pi/2) = -\frac{\cos x}{\sin x}$$
The following is easy