If $x=a(\theta +\sin \theta )$ and $y=a(1-\cos \theta )$, then $\frac{dy}{dx}$ at $\theta =\frac{\pi }{2}$ is

Solution

1. Given $x=a(\theta +\sin \theta )$ and $y=a(1-\cos \theta )$, differentiate both with respect to $\theta$ to get $\frac{dx}{d\theta }=a(1+\cos \theta )$ and $\frac{dy}{d\theta }=a\sin \theta$.
2. Apply the chain rule to find the derivative: $$\begin{array}{rl}\frac{dy}{dx}& =\frac{dy/d\theta }{dx/d\theta }\\ & =\frac{a\sin \theta }{a(1+\cos \theta )}\\ & =\frac{\sin \theta }{1+\cos \theta }\end{array}$$.
3. Substitute $\theta =\frac{\pi }{2}$ into the expression: $$\begin{array}{rl}\frac{dy}{dx}& =\frac{\sin (\pi /2)}{1+\cos (\pi /2)}\\ & =\frac{1}{1+0}\\ & =1\end{array}$$.
4. Hence the answer is (B).