Answer
(a) $x = tan~t$
$y = cot~t$
for $t$ in $(0,\frac{\pi}{2})$
We can see graph the below.
(b) $y = \frac{1}{x}$
for $x$ in $(0,\infty)$
Work Step by Step
(a) $x = tan~t$
$y = cot~t$
for $t$ in $(0,\frac{\pi}{2})$
When $t = \frac{\pi}{10}$:
$x = tan~\frac{\pi}{10} = 0.32$
$y = cot~\frac{\pi}{10} = 3.08$
When $t = \frac{\pi}{6}$:
$x = tan~\frac{\pi}{6} = 0.58$
$y = cot~\frac{\pi}{6} = 1.73$
When $t = \frac{\pi}{4}$:
$x = tan~\frac{\pi}{4} = 1$
$y = cot~\frac{\pi}{4} = 1$
When $t = \frac{\pi}{3}$:
$x = tan~\frac{\pi}{3} = 1.73$
$y = cot~\frac{\pi}{3} = 0.58$
When $t = \frac{2\pi}{5}$:
$x = tan~\frac{2\pi}{5} = 3.08$
$y = cot~\frac{2\pi}{5} = 0.32$
We can see graph the below.
Note that $y=0$ is an asymptote as $x$ goes to infinity and $x=0$ is an asymptote as $x$ approaches 0.
(b) $x = tan~t$
$y = cot~t = \frac{1}{tan~t}$
Therefore: $~~y = \frac{1}{x}$
Since $t$ in $(0,\frac{\pi}{2})$, then $x$ in $(0,\infty)$
