Tangent to Cycloid is Vertical at Cusps

Theorem
The tangent to the cycloid whose locus is given by:
 * $x = a \left({\theta - \sin \theta}\right)$
 * $y = a \left({1 - \cos \theta}\right)$

is vertical at the cusps.

Proof
From Slope of Tangent to Cycloid, the slope of the tangent to $C$ at the point $\left({x, y}\right)$ is given by:
 * $\dfrac {\mathrm d y} {\mathrm d x} = \cot \dfrac \theta 2$

At the cusps, $\theta = 2 n \pi$ for $n \in \Z$.

Thus at the cusps, the slope of the tangent to $C$ is $\cot n \pi$.

From Shape of Cotangent Function:
 * $\displaystyle \lim_{\theta \mathop \to n \pi} \cot \theta \to \infty$

Hence the result by definition of vertical tangent line.