Question

If $y = \tan^{-1}\left(\sqrt{\frac{1+\cos x}{1-\cos x}}\right)$, then $\frac{dy}{dx} =$

• A. $1$
• B. $\frac{3}{2}$
• C. $\frac{1}{2}$
• D. $-\frac{1}{2}$

Hint:

Memorize standard reductions to save precious time: $\sqrt{\frac{1+\cos x}{1-\cos x}} = \cot\left(\frac{x}{2}\right)$. Since $\tan^{-1}(\cot\theta) = \frac{\pi}{2} - \theta$, the function reduces instantly to $\frac{\pi}{2} - \frac{x}{2}$. Its derivative is simply the coefficient of $x$, which is $-\frac{1}{2}$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The given problem requires finding the first derivative of an inverse trigonometric function, $y = \tan^{-1}\left(\sqrt{\frac{1+\cos x}{1-\cos x}}\right)$, with respect to $x$. Differentiating the expression directly using the chain rule would be highly tedious, so we must simplify the trigonometric expression inside the radical first.

Step 2: Key Formula or Approach:
We use the fundamental half-angle trigonometric identities: $$1 + \cos x = 2\cos^2\left(\frac{x}{2}\right)$$ $$1 - \cos x = 2\sin^2\left(\frac{x}{2}\right)$$ Substituting these into the radical simplifies the function to an easily differentiable form.

Step 3: Detailed Explanation:
Let's substitute the half-angle formulas into the given equation: $$y = \tan^{-1}\left(\sqrt{\frac{2\cos^2\left(\frac{x}{2}\right)}{2\sin^2\left(\frac{x}{2}\right)}}\right)$$ The factor of 2 cancels out from the numerator and denominator: $$y = \tan^{-1}\left(\sqrt{\cot^2\left(\frac{x}{2}\right)}\right)$$ Taking the square root converts the expression into a standard cotangent function: $$y = \tan^{-1}\left(\cot\left(\frac{x}{2}\right)\right)$$ To match the outer inverse tangent function, convert the cotangent term into a tangent function using the co-function identity $\cot\theta = \tan\left(\frac{\pi}{2} - \theta\right)$: $$y = \tan^{-1}\left(\tan\left(\frac{\pi}{2} - \frac{x}{2}\right)\right)$$ Using the property $\tan^{-1}(\tan\theta) = \theta$: $$y = \frac{\pi}{2} - \frac{x}{2}$$ Now, differentiate this simplified equation with respect to $x$: $$\frac{dy}{dx} = \frac{d}{dx}\left(\frac{\pi}{2}\right) - \frac{d}{dx}\left(\frac{x}{2}\right)$$ $$\frac{dy}{dx} = 0 - \frac{1}{2} = -\frac{1}{2}$$

Step 4: Final Answer:
The derivative $\frac{dy}{dx}$ is equal to $-\frac{1}{2}$, which matches option (D).