Why is it that the graphs of tan inverse and sin in the interval $$\left[-\frac \pi 2 , \frac \pi 2\right]$$ are so similar.
Is it just some coincidence or something deeper?
It's nothing deep.
Both $f(x) = \arctan x$ and $g(x) = \sin x$ are increasing on $(-\pi/2,\pi/2)$ and both have an inflection point at $x = 0$. Their somewhat similar appearance is due to those properties and the fact that $\arctan \dfrac \pi 2$ is remarkably close to $1$.
The first few terms of their Taylor series around $0$ are $x-x^3/6+x^5/120$ and $x-x^3/3+x^5/5$, respectively, so it’s not terribly surprising that they look similar in a relatively small interval around the origin. I doubt that there’s any deep connection beyond that, though.