Question

If \( y = \sqrt{\frac{1 + \cos 2\theta}{1 - \cos 2\theta}} \), then \( \frac{dy}{d\theta} \) at \( \theta = \frac{3\pi}{4} \) is:

• A. -2
• B. 2
• C. \( \pm 2 \)
• D. None of these

Hint:

When differentiating functions involving trigonometric identities, use the chain and quotient rule appropriately. Make sure to evaluate at the given point to find the final answer.

Answer 1

The Correct Option is A

\[ y = \sqrt{\frac{1 + \cos 2\theta}{1 - \cos 2\theta}} \] \[ \Rightarrow y = \sqrt{\frac{2\cos^2\theta}{2\sin^2\theta}} = \sqrt{\cot^2\theta} \] \[ \Rightarrow y = \cot \theta \] Differentiate w.r.t. \( \theta \), we get: \[ \frac{dy}{d\theta} = -\csc^2\theta \] Now, \[ \left( \frac{dy}{d\theta} \right)_{\theta = \frac{3\pi}{4}} = -\csc^2 \left(\frac{3\pi}{4} \right) \] \[ = -\csc^2 \left(\pi - \frac{\pi}{4} \right) = -\csc^2 \frac{\pi}{4} = -2 \]