Question

If $x=a\left(\theta -\sin \theta \right)andy=a\left(1+\cos \theta \right)$ then prove that $\frac{dy}{dx}=-\cot \left(\frac{\theta }{2}\right)$.

Answer 1

$x=a(\theta -\sin \theta )$

$\frac{dx}{d\theta }=a(1-\cos \theta )$

$y=a(1+\cos \theta )$

$\frac{dy}{d\theta }=a(0-\sin \theta )$

$=-a\sin \theta$

$\therefore \frac{dy}{dx}=\frac{dy/d\theta }{dx/d\theta }=\frac{-a\sin \theta }{a(1-\cos \theta )}$

$=\frac{-\sin \theta }{1-\cos \theta }=\frac{-2\sin \theta /2\cos \theta /2}{2{\sin }^2\theta /2}$

$=-\cot \frac{\theta }{2}$

$\therefore \frac{dy}{dx}=-\cot \frac{\theta }{2}$