The precise task was to determine the finite series of $\arctan x$ with the Lagrange form of the remainder.
While going through the process, I tried to find any pattern in the successive derivatives but could not find any help. Additionally, I found the recursive relation between the derivatives, but that didn't help either.
When I checked the answer in the back, the remainder was in the form of $\sin^{n} \theta$,in fact the remainder was $\: R_n =\frac{(-1)^{n-1}x^n}{n}\sin^n \,(\cot^{-1} \theta x )\sin (\cot^{-1} \theta x )$
I am totally dumbstruck regarding how that sin function is coming. Any hint will be highly appreciated. However I found a similar question that didn't help either.