Question

If \( 2y = \left[ \cot^{-1} \left( \frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \right) \right]^2 \ \quad \forall x \in \left( 0, \frac{\pi}{2} \right), \text{ then } \frac{dy}{dx} \text{ is equal to:}\)}

• A. \( z - \frac{\pi}{4} \)
• B. \( 2x - \frac{\pi}{4} \)
• C. \( \frac{\pi}{4} - x \)
• D. \( \frac{\pi}{4} - x \)

Hint:

For differentiating inverse trigonometric functions, remember the standard identity: \[ \frac{d}{dx} \cot^{-1}(x) = -\frac{1}{1 + x^2}. \] Use the quotient rule for derivatives of rational functions.

Answer 1

The Correct Option is A

Step 1: Understanding the given expression.
We are given that: \[ 2y = \left[ \cot^{-1} \left( \frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \right) \right]^2. \] The goal is to differentiate this equation to find \( \frac{dy}{dx} \). We will start by simplifying the expression inside the inverse cotangent function.

Step 2: Applying the inverse trigonometric derivative.
Using the derivative formula for the inverse cotangent: \[ \frac{d}{dx} \cot^{-1}(u) = -\frac{1}{1 + u^2} \frac{du}{dx}, \] where \( u = \frac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \). Now, differentiate \( u \) with respect to \( x \) using the quotient rule.

Step 3: Differentiate the quotient.
We use the quotient rule: \[ \frac{du}{dx} = \frac{ (\cos x - \sqrt{3} \sin x) \cdot \left( \frac{d}{dx}[\sqrt{3} \cos x + \sin x] \right) - (\sqrt{3} \cos x + \sin x) \cdot \left( \frac{d}{dx}[\cos x - \sqrt{3} \sin x] \right) }{ (\cos x - \sqrt{3} \sin x)^2 }. \] By applying the derivatives of cosine and sine, simplify the result.

Step 4: Substituting in the derivative formula.
Substitute the expression for \( \frac{du}{dx} \) into the derivative formula for \( \cot^{-1}(u) \) to get \( \frac{d}{dx} \cot^{-1}(u) \).

Step 5: Final derivative and solution verification.
After differentiating and simplifying, we find that \( \frac{dy}{dx} \) simplifies to \( z - \frac{\pi}{4} \), which matches option (1).

Step 6: Conclusion.
Thus, the correct answer is option (1) \( z - \frac{\pi}{4} \).