I need to calculate the value of residue of $f(z) = \tan^2z$ at $z_k=\frac{\pi}{2} + \pi k$. I know that these points are poles of the second order, so the common way is to calculate residue by the formula.
$$
Res_{z_k}(\tan^2z) = \lim_{z\to z_k}\frac{d}{dz}((z-z_k)^2\cdot\tan^2(z))
$$
But this limit is very unpleasant to calculate, at least for me. Maybe I am missing something?
The other way is to calculate the Laurent series at $z_k$. And the only way I see to do so, is by dividing the squares of the Taylor series of sine and cosine, which is also a hell of a thing to do.
The residue of $\tan(z)^2$ at $\frac{\pi}{2} + \pi k$ is equal to the residue of $$ z \mapsto \tan(z - \frac{\pi}{2} - \pi k)^2 $$ at $z=0$. But that is an even function, so that its Laurent series at $z=0$ has only terms with even exponents. In particular, the residue at $z=0$ is zero.