Question

If \[ y = \tan^{-1}\left( \frac{\sqrt{1 + \cos \frac{x}{2}}}{\sqrt{1 - \cos \frac{x}{2}}} \right) \] then \( \frac{dy}{dx} \) is

• A. \( -\frac{1}{3} \)
• B. \( -\frac{1}{4} \)
• C. \( \frac{1}{3} \)
• D. \( \frac{1}{4} \)

Hint:

Always simplify expressions using trigonometric identities before differentiating.

Answer 1

The Correct Option is B

Step 1: Simplify the given expression.
We start by simplifying the given expression inside the inverse tangent function. Using the identity \( \frac{1 + \cos \theta}{1 - \cos \theta} = \tan^2 \left( \frac{\theta}{2} \right) \), we rewrite the expression inside the tangent inverse: \[ \frac{\sqrt{1 + \cos \frac{x}{2}}}{\sqrt{1 - \cos \frac{x}{2}}} = \tan \left( \frac{x}{4} \right) \] Step 2: Take the derivative.
Now we differentiate \( y = \tan^{-1} \left( \tan \frac{x}{4} \right) \). We know that the derivative of \( \tan^{-1} (z) \) is \( \frac{1}{1+z^2} \), so \[ \frac{dy}{dx} = \frac{1}{1 + \left( \tan \frac{x}{4} \right)^2} \cdot \frac{d}{dx} \left( \frac{x}{4} \right) \] Step 3: Simplify the result.
\[ \frac{dy}{dx} = \frac{1}{1 + \tan^2 \frac{x}{4}} \cdot \frac{1}{4} \] Using the identity \( 1 + \tan^2 \theta = \sec^2 \theta \), we get \[ \frac{dy}{dx} = \frac{1}{4 \sec^2 \frac{x}{4}} = \frac{1}{4} \] Step 4: Conclusion.
The required derivative is \( \frac{dy}{dx} = -\frac{1}{4} \).