The derivative \(\displaystyle\frac{dy}{dx}\), if \(x = a(\theta - \sin\theta)\), \(y = a(1 + \cos\theta)\) is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(-\cot\displaystyle\frac{\theta}{2}\)
For a curve in parametric form, \(\frac{dy}{dx} = \frac{dy/d\theta}{dx/d\theta}\).
\( \displaystyle \frac{dx}{d\theta} = a(1 - \cos\theta) \)
\( \displaystyle \frac{dy}{d\theta} = -a\sin\theta \)
So
\( \displaystyle \frac{dy}{dx} = \frac{-\sin\theta}{1 - \cos\theta} \)
Using the half-angle formulas \(\sin\theta = 2\sin\frac{\theta}{2}\cos\frac{\theta}{2}\) and \(1 - \cos\theta = 2\sin^2\frac{\theta}{2}\):
\( \displaystyle \frac{dy}{dx} = \frac{-2\sin\frac{\theta}{2}\cos\frac{\theta}{2}}{2\sin^2\frac{\theta}{2}} \)
\( \displaystyle = -\cot\frac{\theta}{2} \)
NCERT (2026-27 edition):
Derivatives of Functions in Parametric Forms
