# Approximation of tan(f(n))

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 $$\infty$$

• Your function is $f(x)$, but it is written in terms of $n$. Should it be $f(n)$? – Calvin Godfrey Jan 11 at 13:28
• What if $a\approx2\pi(n+1/2)$? – Barry Cipra Jan 11 at 13:37

The first step is to use

$$\tan(x +n\pi +\pi/2) = -\frac{\cos x}{\sin x}$$

The following is easy