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AI-তৈরি নমুনা উত্তর

Given,
$x = a(\theta + \sin\theta) \implies \frac{dx}{d\theta} = a(1 + \cos\theta) = 2a\cos^2\left(\frac{\theta}{2}\right)$
$y = a(1 - \cos\theta) \implies \frac{dy}{d\theta} = a\sin\theta = 2a\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)$

Therefore,
$\frac{dy}{dx} = \frac{\frac{dy}{d\theta}}{\frac{dx}{d\theta}} = \frac{2a\sin\left(\frac{\theta}{2}\right)\cos\left(\frac{\theta}{2}\right)}{2a\cos^2\left(\frac{\theta}{2}\right)} = \tan\left(\frac{\theta}{2}\right)$

Differentiating with respect to $x$:
$\frac{d^2y}{dx^2} = \frac{d}{dx}\left[\tan\left(\frac{\theta}{2}\right)\right] = \frac{d}{d\theta}\left[\tan\left(\frac{\theta}{2}\right)\right] \cdot \frac{d\theta}{dx}$
$\frac{d^2y}{dx^2} = \frac{1}{2}\sec^2\left(\frac{\theta}{2}\right) \cdot \frac{1}{2a\cos^2\left(\frac{\theta}{2}\right)} = \frac{1}{4a}\sec^4\left(\frac{\theta}{2}\right)$