If the parametric equation of a curve is given by $x=e^t \cot t, y=e^t$, then the tangent to the curve at the point $t=\frac{\pi}{4}$ makes with the axis of x-axis the angle

$\frac{\pi}{2}$

$\frac{d y}{d x}=\frac{\frac{d y}{d t}}{\frac{d x}{d t}}=\frac{e^t(\sin t+\cos t)}{e^t \cos t-\sin t}=\infty$  at  $t=\frac{\pi}{4}$

∴ the tangent is perpendicular to x-axis.