Tangent to Cycloid

Theorem
Let $C$ be a cycloid generated by the equations:
 * $x = a \left({\theta - \sin \theta}\right)$
 * $y = a \left({1 - \cos \theta}\right)$

Then the tangent to $C$ at a point $\left({x, y}\right)$ on $C$ is given by the equation:
 * $y - a \left({1 - \cos \theta}\right) = \dfrac {\sin \theta} {1 - \cos \theta} \left({x - a \theta + a \sin \theta}\right)$

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

This tangent to $C$ also passes through the point $\left({a \left({\theta - \sin \theta}\right), a \left({1 - \cos \theta}\right)}\right)$.

Hence the result.