If $x = 2(t - \sin t)$ and $y = 2(1 - \cos t)$, what is $\frac{dy}{dx}$?

Explanation

1. 01

   $\frac{dx}{dt} = 2(1 - \cos t)$ and $\frac{dy}{dt} = 2\sin t$, so $\frac{dy}{dx} = \frac{\sin t}{1 - \cos t}$.

2. 02

   With $\sin t = 2\sin\frac{t}{2}\cos\frac{t}{2}$ and $1 - \cos t = 2\sin^2\frac{t}{2}$, this is $\cot\frac{t}{2}$.