Find $\frac{dy}{dx}$, if $x = a(\theta + \sin \theta), y = a(1 - \cos \theta)$.

$\tan\left(\frac{\theta}{2}\right)$

The correct answer is Option (3) → $\tan\left(\frac{\theta}{2}\right)$ ##

We have

$\frac{dx}{d\theta} = a(1 + \cos \theta), \frac{dy}{d\theta} = a(\sin \theta)$

Therefore $\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{a \sin \theta}{a(1 + \cos \theta)} = \tan \frac{\theta}{2}$