Starting from \begin{equation*} \frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots, \end{equation*} which holds when $|x| < 1$, we can conclude that \begin{equation*} \frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \cdots \end{equation*} when $|x| < 1$. Now taking antiderivatives of both sides, we discover that \begin{equation*} \tan^{-1}(x) = C + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \cdots \end{equation*} when $|x| < 1$. By plugging in $x = 0$, we see that $C = 0$, and we have found the Taylor series (centered at $0$) for $\tan^{-1}$.
This is a pretty neat calculation. Computing the derivatives of $\tan^{-1}$ at $0$ directly would have been laborious. What are some other neat or good examples of calculations like this (where a Taylor series is computed in an efficient way)?
A couple other examples would be computing the Maclaurin series for $x \sin(x)$ by just multiplying the Maclaurin series for $\sin(x)$ by $x$, or computing the Maclaurin series for $\sin(x) \cos(x)$ by multiplying the series for $\sin(x)$ by the series for $\cos(x)$. (You could also use the identity $\sin(x) \cos(x) = \frac12 \sin(2x)$.)