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If $x$ and $y$ are connected parametrically by the given equations, without eliminating the parameter. Find $\frac{d y}{d x}$
$x=a(\theta-\sin \theta), y=a(1+\cos \theta)$
$x=a(\theta-\sin \theta), y=a(1+\cos \theta)$
Solution:
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Verified Answer
$\frac{d x}{d \theta}=a[1-\cos \theta] \& \frac{d y}{d \theta}=-a \sin \theta$
$\frac{d y}{d x}=\frac{-a \sin \theta}{a(1-\cos \theta)}=\frac{-2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}}{2 \sin ^2 \frac{\theta}{2}}=-\cot \frac{\theta}{2}$
$\frac{d y}{d x}=\frac{-a \sin \theta}{a(1-\cos \theta)}=\frac{-2 \sin \frac{\theta}{2} \cdot \cos \frac{\theta}{2}}{2 \sin ^2 \frac{\theta}{2}}=-\cot \frac{\theta}{2}$
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