Tangent to Cycloid

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

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

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

This tangent to $C$ also passes through the point $\tuple {a \paren {\theta - \sin \theta}, a \paren {1 - \cos \theta} }$.

Hence the result.