Compute the first four non-zero terms in the Taylor series for the function $\arctan(x^2 -1)$ centred about the point $a = 0$.
I think there is easy way to solve this, starting from $$\arctan x \approx x - \frac 1 3 x^3 + \frac 1 5 x^5 + \dots $$ instead of differentiating step by step, which is a lot of work.
But when I substitute $x^2 -1$ into the Taylor series for $\arctan x$, the answer is different from the correct answer and I'm not sure why. Can't we find a Taylor series by making use of an existing Taylor series?