How to expand $\tan z$ at $z_{0}= \frac{\pi}{4}$, is it a concise form？
Solution 1:
The series expansion you are looking for is of the form:
$$\sum_{n=0}^\infty a_n\left(z-\frac{π}{4}\right)^n$$
with
$$a_n=\frac{f^{(n)}\left(\frac{π}{4}\right)}{n!}\;,$$
where $f^{(n)}\left(\frac{π}{4}\right)$ denotes the n-th derivative of $f(z)=\tan z$ calculated at $z=\frac{π}{4}\;.$
With the above reminder of Taylor series theory you can now write a few terms of the expansion.