Recall that [tex]x=r\cos\theta[/tex]. When [tex]\cos\theta\neq0[/tex], we can write
[tex]r=\sin\theta\tan\theta\iff r\cos\theta=x=\sin^2\theta[/tex]
You have [tex]x=1[/tex] whenever [tex]\theta[/tex] is a multiple of [tex]\dfrac\pi2[/tex]. But [tex]\cos\dfrac\pi2=0[/tex], which would make [tex]r[/tex] undefined. So [tex]x=1[/tex] must be a vertical asymptote.
So assume [tex]\theta\neq\dfrac{n\pi2}[/tex] for any integer [tex]n[/tex]. Then
[tex]x=\sin^2\theta[/tex]
and since [tex]|\sin\theta|<1[/tex] for all [tex]\theta[/tex], it follows that [tex]0<\sin^2\theta<1[/tex], or [tex]0<x<1[/tex].
