Question

If $u=\sec^{-1}(-\sec 2\theta)$ and $v=\cos \theta$, then $\frac{du}{dv}$ at $\theta=\frac{\pi}{4}$ is equal to:

• A. $\sqrt{2}$
• B. $2\sqrt{2}$
• C. $\frac{1}{\sqrt{2}}$
• D. $\frac{1}{2\sqrt{2}}$
• E. $-\sqrt{2}$

Hint:

Simplify the inverse trigonometric expression using identities before differentiating to save time.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We need to find the derivative of one function (u) with respect to another function (v). This is a parametric derivative problem, where both u and v are functions of a third variable, \( \theta \). The goal is to find \( \frac{du}{dv} \).
Step 2: Key Formula or Approach:
1. Parametric Differentiation: \( \frac{du}{dv} = \frac{du/d\theta}{dv/d\theta} \).
2. Simplify the expression for u first. We use the identity \( \sec(\pi - x) = -\sec x \). So, \( \sec^{-1}(-\sec 2\theta) = \pi - \sec^{-1}(\sec 2\theta) = \pi - 2\theta \), assuming \( 2\theta \) is in the appropriate range. For \( \theta=\pi/6 \), \( 2\theta=\pi/3 \) which is in \( [0, \pi/2) \), so the simplification \( \sec^{-1}(\sec 2\theta)=2\theta \) is valid.
So, \( u = \pi - 2\theta \).
Step 3: Detailed Explanation:
Let's simplify u and v and find their derivatives with respect to \( \theta \).
Function u:
\[ u = \sec^{-1}(-\sec 2\theta) = \pi - 2\theta \] Derivative of u with respect to \( \theta \):
\[ \frac{du}{d\theta} = \frac{d}{d\theta}(\pi - 2\theta) = -2 \] Function v:
\[ v = \cos^2 \theta \] Derivative of v with respect to \( \theta \) (using the chain rule):
\[ \frac{dv}{d\theta} = 2(\cos \theta)^1 \cdot \frac{d}{d\theta}(\cos \theta) = 2\cos \theta (-\sin \theta) = -\sin(2\theta) \] Now, find \( \frac{du}{dv} \) using the parametric derivative formula:
\[ \frac{du}{dv} = \frac{du/d\theta}{dv/d\theta} = \frac{-2}{-\sin(2\theta)} = \frac{2}{\sin(2\theta)} \] We need to evaluate this derivative at \( \theta = \frac{\pi}{6} \).
\[ \left. \frac{du}{dv} \right|_{\theta=\pi/6} = \frac{2}{\sin(2 \cdot \frac{\pi}{6})} = \frac{2}{\sin(\frac{\pi}{3})} \] We know that \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).
\[ \frac{du}{dv} = \frac{2}{\sqrt{3}/2} = \frac{4}{\sqrt{3}} \] This result does not match the provided answer key or any of the options. There is a definite error in the question or the provided answer key. Let's re-examine the OCR for `v`. `v = cos^2 \theta`. What if it was `v = cos \theta^2`? No, that's less likely. What if it was `v = \cos 2\theta`? If \( v = \cos 2\theta \), then \( \frac{dv}{d\theta} = -2 \sin 2\theta \). Then \( \frac{du}{dv} = \frac{-2}{-2 \sin 2\theta} = \frac{1}{\sin 2\theta} \). At \( \theta=\pi/6 \), this is \( \frac{1}{\sin(\pi/3)} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}} \). Still no match. Given the discrepancy, the problem is flawed. No standard interpretation leads to the given answer. Step 4: Final Answer:
Based on a standard interpretation of the question, the derivative is \( \frac{4}{\sqrt{3}} \). This does not match any option. The question is likely erroneous.