[tex]\dfrac{\mathrm dy}{\mathrm dx}=\dfrac{\frac{\mathrm dy}{\mathrm dt}}{\frac{\mathrm dx}{\mathrm dt}}[/tex]
[tex]\dfrac{\mathrm dy}{\mathrm dt}=4\pi\cos\left(4\pi t+\dfrac\pi2\right)[/tex]
[tex]\dfrac{\mathrm dx}{\mathrm dt}=\pi-4\pi\sin\left(4\pi t+\dfrac\pi2\right)[/tex]
So the slope at [tex]t[/tex] is
[tex]\dfrac{4\pi\cos\left(4\pi t+\dfrac\pi2\right)}{\pi-4\pi\sin\left(4\pi t+\dfrac\pi2\right)}=\dfrac{-4\sin(4\pi t)}{1-4\cos(4\pi t)}[/tex]
The tangent line will be vertical when [tex]\dfrac{\mathrm dx}{\mathrm dt}=0[/tex], which occurs for
[tex]\pi-3\pi\sin\left(4\pi t+\dfrac\pi2\right)=\pi-4\pi\cos(4\pi t)=0\implies\cos(4\pi t)=\dfrac14\implies t=\pm\dfrac{\arccos\frac14}{4\pi}+\dfrac n2[/tex]
where [tex]n[/tex] is an integer. Assuming [tex]t>0[/tex], we get [tex]\dfrac{\mathrm dx}{\mathrm dt}=0[/tex] when [tex]t=\dfrac{\arccos\frac14}{4\pi}+\dfrac n2[/tex]. So the first time the tangent line will be vertical is when [tex]n=0[/tex], and the second when [tex]n=1[/tex]. In other words,
[tex]t_1=\dfrac1{4\pi}\arccos\dfrac14\approx0.1049[/tex]
[tex]t_2=\dfrac1{4\pi}\arccos\dfrac14+\dfrac12\approx0.6049[/tex]
