The tangent line is horizontal whenever [tex]\dfrac{\mathrm dy}{\mathrm d\theta}=0[/tex].
[tex]\dfrac{\mathrm dy}{\mathrm d\theta}=2\cos2\theta=0\implies2\theta=\dfrac{(2n+1)\pi}2\implies\theta=\dfrac{(2n+1)\pi}4[/tex]
where [tex]n[/tex] is any integer.
I'm guessing you're only interested in one complete loop of the lemniscate. In that case, we're restricted to [tex]0\le\theta\le2\pi[/tex]. Then we get 4 points of interest for [tex]n=0,1,2,3[/tex]:
[tex]\theta=\dfrac\pi4,\dfrac{3\pi}4,\dfrac{5\pi}4,\dfrac{7\pi}4[/tex]
