[tex]f(x)=\arctan x[/tex]
[tex]\dfrac{\mathrm df}{\mathrm dx}=\dfrac1{1+x^2}[/tex]
[tex]\dfrac{\mathrm d^2f}{\mathrm dx^2}=-\dfrac{2x}{(1+x^2)^2}[/tex]
[tex]\dfrac{\mathrm d^3f}{\mathrm dx^3}=\dfrac{6x^2-2}{(1+x^2)^3}[/tex]
[tex]\dfrac{\mathrm d^4f}{\mathrm dx^4}=\dfrac{24x-24x^3}{(1+x^2)^4}[/tex]
The Taylor polynomial takes the form
[tex]P_4(x)=f(0)+\dfrac{f'(0)}{1!}(x-0)+\dfrac{f''(0)}{2!}(x-0)^2+\dfrac{f'''(0)}{3!}(x-0)^3+\dfrac{f^{(4)}(0)}{4!}(x-0)^4[/tex]
[tex]P_4(x)=\dfrac11x-\dfrac26x^3=x-\dfrac13x^3[/tex]
Notice that the fourth degree term vanishes, since [tex]f^{(4)}(0)=0[/tex], so you can stop at the third degree.
