ndf[θ_,α_]:=((α^2)/(π*(((((Cos[θ])^2)*((α^2)-1)+1))^2)))
p[μ_,α_]:=((α^2)/(π((((μ^2)((α^2)-1)+1))^2)))μ

Manipulate[ParametricPlot[p[μ,α]*{(Sqrt[1-(μ^2)]),μ},{μ,1,-1},PlotRange->{{-1,1},{0,4}}],{{α,0.1},0,1}]
Manipulate[ParametricPlot[ndf[θ,α]*Cos[θ]*{Sin[θ],Cos[θ]},{θ,-(π/2),(π/2)},PlotRange->{{-1,1},{0,4}}],{{α,0.1},0,1}]

Manipulate[Plot[p[μ,α],{μ,0,1},PlotRange->Full],{{α,0.1},0,1}]

Clear[μ,θ,α]
Integrate[p[μ,α]*(Sqrt[1-(μ^2)]),μ]

Integrate[ndf[θ,α]*Cos[θ]*Sin[θ],θ]

cdf[θ_,α_]:=((α^2)/(2 π (-1+(α^2)) (1+(-1+(α^2)) ((Cos[θ])^2))))
Manipulate[Plot[2π *(cdf[θ,α]-cdf[0,α]),{θ,0,(π/2)},PlotRange->{0,1},AxesLabel->{"θ","cdf"}],{{α,0.1},0,1}]