Laurent expansion of the tangent function about a given point

Question

I am given the following question :

Show that if $\tan z$ is expanded into a Laurent series about $z = \pi/2$,

(a) the principal part is $-1/(z - \pi/2)$,

(b) the series converges for $0 < |z - \pi/2| < \pi/2$,

(c) $z = \pi/2$ is a simple pole.

Once parts (a) and (b) are solved, part (c) will follow immediately, but I am not getting any idea on how to expand the function about the given point.

Thanks in advance.

Answer 1

By using a property of the $\tan$ function, all we need is its expansion at $0$.

Let $w=z-\pi/2$, then $$\tan(z)=\tan(w+\pi/2)=-\frac{1}{\tan(w)}=-\frac{1}{w+O(w^3)}=-\frac{1}{w}\cdot \frac{1}{1+O(w^2)}.$$ Can you take it from here?