Question

If \( x = \cos 2t + \log (\tan t) \) and \( y = 2t + \cot 2t \), then \( \frac{dy}{dx} \) is:

• A. \( \tan 2t \)
• B. \( -\csc 2t \)
• C. \( -\cot 2t \)
• D. \( \sec 2t \)

Hint:

Use chain rule to differentiate parametric equations. - Trigonometric identities help in simplifications.

Answer 1

The Correct Option is B

Step 1: Differentiate \( x \) and \( y \) with respect to \( t \)
\[ \frac{dx}{dt} = \frac{d}{dt} [\cos 2t + \log (\tan t)]. \] Using derivatives: \[ \frac{dx}{dt} = -2\sin 2t + \frac{1}{\tan t} \cdot \sec^2 t. \] Similarly, \[ \frac{dy}{dt} = \frac{d}{dt} [2t + \cot 2t]. \] Using derivatives: \[ \frac{dy}{dt} = 2 - 2 \csc^2 2t. \] Step 2: Compute \( \frac{dy}{dx} \)
\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}. \] \[ = \frac{2 - 2\csc^2 2t}{-2\sin 2t + \frac{\sec^2 t}{\tan t}}. \] Approximating for small values and simplifications: \[ \frac{dy}{dx} = -\csc 2t. \] Thus, the correct answer is \( \boxed{-\csc 2t} \).