x=cos(θ),
y=cos(3θ)=
cos(2θ)cos(θ)-sin(2θ)sin(θ)=
2cos3(θ)-cos(θ)-2sin2(θ)cos(θ)=
2cos3(θ)-cos(θ)-2cos(θ)+2cos3(θ)=
4cos3(θ)-3cos(θ)=4x3-3x
dy/dx=12x2-3=0 when tangent is horizontal, x2=¼, x=±½.
y=1/2-3/2=-1 or -1/2+3/2=1.
So the points are (½,-1) and (-½,1).