f=Normal[Series[(E^x),{x,0,9}]]

g=Normal[Series[Cos[x],{x,0,10}]]

Expand[f*g]

Normal[Series[(E^x)Cos[x],{x,0,10}]]

Manipulate[
Show[
Plot[
{(E^x)Cos[x],Evaluate[Normal[Series[(E^x)Cos[x],{x,x0,n}]]]},{x,-3,3},
PlotRange->{{-3,3},{-10,3}},
PlotStyle->{{RGBColor[0, 0.501961, 1],Thickness[0.010]},{Thick,Orange}},
ImageSize->600,
AspectRatio->0.8,
LabelStyle->(FontSize->12)
],
ListPlot[
{{x0,(E^x0) Cos[x0]}},
PlotStyle->{Black,PointSize[0.015]}
]
],
{{n,0,"n"},0,20,1, Appearance->"Labeled"},
{{x0,0,"(Subscript[x, 0])"},-3,3,0.1, Appearance->"Labeled"}
]

Manipulate[
Show[
Plot[
{Sin[θ],Evaluate[Normal[Series[Sin[θ],{θ,θ0,n}]]]},{θ,-π,π},
PlotRange->{{-π,π},{-1.1,1.1}},
PlotStyle->{{RGBColor[0, 0.501961, 1],Thickness[0.010]},{Thick,Orange}},
ImageSize->600,
AspectRatio->0.8,
LabelStyle->(FontSize->12)
],
ListPlot[
{{θ0,Sin[θ0]}},
PlotStyle->{Black,PointSize[0.015]}
]
],
{{n,0,"n"},0,41,1, Appearance->"Labeled"},
{{θ0,0,"(Subscript[θ, 0])"},-π,π,(π/12), Appearance->"Labeled"}
]