Looks good to me. If I was going to offer a critique I would just say: when writing an argument it's always better to over communicate rather than under communicate.
The first equality is just algebra.
Your second equality requires a little bit to see clearly but it's true. Most will recall:
$$\tan(A+B)=\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)}$$
Or if you'd like:
$$A+B= \tan^{-1} \bigg(\frac{\tan(A)+\tan(B)}{1-\tan(A)\tan(B)} \bigg)$$
Taking $A=\tan^{-1}(1+n)$ and $B=\tan^{-1}(1-n)$
Honestly adding this much explanation seems like almost overkill.
The 4th equality follows as result of $\tan^{-1}$ being an odd function.
Now the last part you are using a telescoping series technique so that you may ignore all the middle terms. That is,
$$\begin{align} &\sum_{n=1}^\infty\tan^{-1}(n+1)-\tan^{-1}(n-1) \\
&= \lim_{m\to \infty} \tan^{-1}(m+1)-tan^{-1}(m-1)+\dots +\tan^{-1}(4)-\tan^{-1}(2)+\tan^{-1}(3)-\tan^{-1}(1)+tan^{-1}(2)-\tan^{-1}(0)
\end{align}$$
So after we consider what cancels and what doesn't we find that we only need to concern ourselves with $$\lim_{m\to \infty}\tan^{-1}(m+1)+\tan^{-1}(m-1)-\tan^{-1}(1)$$
So while that is true: I think it might merit a sentence or two just to make sure the audience is following.