Question

If \[ x = 3 \left[ \sin t - \log \left( \cot \frac{t}{2} \right) \right], \quad y = 6 \left[ \cos t + \log \left( \tan \frac{t}{2} \right) \right] \] then find \( \frac{dy}{dx} \).

• A. \( \frac{2\sin^2 t}{1 + \sin t \cos t} \)
• B. \( \frac{2\cos^2 t}{1 + \sin 2t} \)
• C. \( \frac{2\cos^2 t}{1 + \sin t \cos t} \)
• D. \( \frac{1 + \cos 2t}{1 + \sin 2t} \)

Hint:

For parametric differentiation, use chain rule: \( \frac{dy}{dx} = \frac{dy/dt}{dx/dt} \) and simplify using trigonometric identities.

Answer 1

The Correct Option is C

Step 1: Differentiating parametric equations We use: \[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] Differentiating \( x(t) \) and \( y(t) \), simplifying the expression, and substituting known identities yield: \[ \frac{dy}{dx} = \frac{2\cos^2 t}{1 + \sin t \cos t}. \]