Q119 / 180 · COMEDK 2025 (10 May Shift 2) · Mathematics
If $\quad 2 y=\left[\cot ^{-1}\left(\dfrac{\sqrt{3} \cos x+\sin x}{\cos x-\sqrt{3} \sin x}\right)\right]^2 \forall x \in\left(0, \dfrac{\pi}{2}\right)$ then $\dfrac{d y}{d x}$ is equal to
- Correct answerA$x-\dfrac{\pi}{6}$
- B$\dfrac{\pi}{3}-x$
- C$2 x-\dfrac{\pi}{3}$
- D$\dfrac{\pi}{6}-x$
Solution
The given expression is $2y = \left[ \cot^{-1} \left( \dfrac{\sqrt{3} \cos x + \sin x}{\cos x - \sqrt{3} \sin x} \right) \right]^2$.
Divide the numerator and denominator inside the $\cot^{-1}$ function by $\cos x$:
$\dfrac{\sqrt{3} + \tan x}{1 - \sqrt{3} \tan x} = \tan \left( \dfrac{\pi}{3} + x \right)$.
Thus, the expression becomes $\cot^{-1} \left( \tan \left( \dfrac{\pi}{3} + x \right) \right)$.
Using the identity $\cot^{-1} (\tan \theta) = \dfrac{\pi}{2} - \theta$, we have:
$\cot^{-1} \left( \tan \left( \dfrac{\pi}{3} + x \right) \right) = \dfrac{\pi}{2} - \left( \dfrac{\pi}{3} + x \right) = \dfrac{\pi}{6} - x$.
Substituting this back into the equation for $2y$:
$2y = \left( \dfrac{\pi}{6} - x \right)^2$.
Differentiating both sides with respect to $x$:
$2 \dfrac{dy}{dx} = 2 \left( \dfrac{\pi}{6} - x \right) \cdot (-1)$.
$\dfrac{dy}{dx} = - \left( \dfrac{\pi}{6} - x \right) = x - \dfrac{\pi}{6}$.
Answer: $x-\dfrac{\pi}{6}$
Divide the numerator and denominator inside the $\cot^{-1}$ function by $\cos x$:
$\dfrac{\sqrt{3} + \tan x}{1 - \sqrt{3} \tan x} = \tan \left( \dfrac{\pi}{3} + x \right)$.
Thus, the expression becomes $\cot^{-1} \left( \tan \left( \dfrac{\pi}{3} + x \right) \right)$.
Using the identity $\cot^{-1} (\tan \theta) = \dfrac{\pi}{2} - \theta$, we have:
$\cot^{-1} \left( \tan \left( \dfrac{\pi}{3} + x \right) \right) = \dfrac{\pi}{2} - \left( \dfrac{\pi}{3} + x \right) = \dfrac{\pi}{6} - x$.
Substituting this back into the equation for $2y$:
$2y = \left( \dfrac{\pi}{6} - x \right)^2$.
Differentiating both sides with respect to $x$:
$2 \dfrac{dy}{dx} = 2 \left( \dfrac{\pi}{6} - x \right) \cdot (-1)$.
$\dfrac{dy}{dx} = - \left( \dfrac{\pi}{6} - x \right) = x - \dfrac{\pi}{6}$.
Answer: $x-\dfrac{\pi}{6}$
Hence, the correct answer is option: 'A'.