Question

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

• A. 1
• B. 0
• C. -1
• D. 2

Hint:

When \( y = \tan^{-1} (\tan x) \), the derivative is simply 1, since \( \tan^{-1} (\tan x) = x \).

Answer 1

The Correct Option is A

Step 1: Simplifying the expression.
We use the identity for the tangent of a double angle: \[ \frac{\sin 2x}{1 + \cos 2x} = \tan x \] Thus, the equation becomes: \[ y = \tan^{-1} (\tan x) = x \]
Step 2: Derivative of \( y \).
Since \( y = x \), the derivative of \( y \) with respect to \( x \) is: \[ \frac{dy}{dx} = 1 \]
Step 3: Conclusion.
Thus, the value of \( \frac{dy}{dx} \) is 1, corresponding to option (A).