Uniform convergence of specific sequence on a compact subset of the complex plane.

Question

I would like to show the uniform convergence of two sequences $(f_n)$ and $(g_n)$ to $f : z \mapsto \frac{2z}{z^2 - \pi^2}$ and to $g: z \mapsto z$ on any compact K of the complex plane where the functions are defined. Here $$ f_n(z) =\frac{2n \tan \frac{z}{n}}{(\cos^2 \frac{\pi}{n})(n \tan \frac{z}{n})^2 -(n \sin \frac{\pi}{n} )^2} $$ and $$ g_n(z)=n\tan \frac{z}{n}. $$ I can show the simple convergence but I does not know the methods for uniform convergence on compact subsets of the complex plane.