The given parametric equations are[tex] x = csc(t) [/tex] and [tex] y = cot(t) [/tex].
Now, we know that [tex] csc(t)=\frac{1}{sin(t)} [/tex] and [tex] cot(t)=\frac{cos(t)}{sin(t)} [/tex]. A nice look at the given parametric equations tells that:
[tex] x^2-y^2=csc^2(t)-cot^2(t)=\frac{1}{sin^2(t)}-\frac{cos^2(t)}{sin^2(t)}=\frac{1-cos^2(t)}{sin^2(t)}=\frac{sin^2(t)}{sin^2(t)}=1 [/tex]
This is the equation of a Hyperbola.
Thus, the conic section which is drawn by the parametric equations x = csct and y = cott is a hyperbola.
