Trivial expansion

$\begin{aligned} f(x) & = \tan^{-1} \left(\color{#3D99F6} \sin (\pi + x^2) \right) & \small \color{#3D99F6} \text{Note that }\sin (\pi - \theta) = \sin \theta \\ & = \tan^{-1} \left(\color{#3D99F6} \sin (- x^2) \right) \\ & = - \tan^{-1} \left(\sin x^2 \right) & \small \color{#3D99F6} \text{By Maclaurin series} \\ & = - \sin x^2 + \frac {\sin^3 x^2}3 - \frac {\sin^5 x^2}5 + \cdots \end{aligned}$

By Maclaurin series again, $\sin x^2 = x^2 - \dfrac {x^6}{3!} + \dfrac {x^{10}}{5!} + \cdots$ .. Since there is no odd powers of $x$ in $\sin x^2$ , there is no odd powers of $x$ in $f(x)$ or all the coefficients of odd powers of $x$ including $x^{19}$ are $\boxed 0$ .

Two (basically equivalent) ways to do this:

• whatever the expansion of $\arctan \left( \sin(\pi+y) \right)$ is, when we substitute in $y=x^2$ , all of the terms will be in even powers of $x$

• the function is clearly even; so its Maclaurin series can have no odd powers

Either way, the coefficient of $x^{19}$ is zero .

$-x^2+\frac{x^6}{2}-\frac{3 x^{10}}{8}+\frac{83 x^{14}}{240}-\frac{8375 x^{18}}{24192}+\frac{147017 x^{22}}{403200}-\frac{1811857 x^{26}}{4561920}+\frac{193146402541 x^{30}}{435891456000}-\frac{78003909463 x^{34}}{154983628800}+\frac{25248145664891 x^{38}}{43553562624000}+O\left(x^{41}\right)$ .

There is no $x^{19}$ term in the series. Therefore, the coefficient is 0.