Tangent even derivative How to show that $\tan^{(2n)}(0)=0$ for each $n \in \mathbb{N}$ ? I can't quite see the inductive step (or maybe there's another way to do it).
Prove by induction that every even derivative of tangent is a linear combinaion of odd powers of tangent.
You can use the Taylor series for $\tan{x}$. This is $$\tan{x} = \sum_{k=1}^{\infty} \frac{U_{2n+1}x^{2n+1}}{(2n+1)!} = x - \frac{x^3}{6} + O(x^5)$$