Compute $\oint_{|z-\frac{\pi}{2}|=\pi+1}z\cdot \tan(z)dz$
My solution: the integrand is a meromorphic function with simple poles at points: $\frac{\pi}{2}+n\pi$, with $n$ integer. Among these points, $\pm \frac{\pi}{2},\frac{3}{2}\pi$ lie inside the contour. I use the formula for simple poles: $Res\left(\frac{f}{g},z_0\right)=\frac{f(z_0)}{g'(z_0)}$ In my case ($f=z\sin z, g=\cos z$) i get: $Res\left(z\cdot \tan(z),\pm\frac{\pi}{2}\right)=\mp\frac{\pi}{2}$ and $Res\left(z\cdot \tan(z),\frac{3\pi}{2}\right)=-\frac{3\pi}{2}$ I apply residue formula to get $I=-3\pi^2i$. Someone could say me if i made some mistakes?