Question

Differentiate \( \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \) with respect to \( x \).

Hint:

Expressions like \( \frac{1-x^2}{1+x^2} \) often relate to \( \cos(2\tan^{-1}x) \). Use identities to simplify inverse trig problems.

Answer 1

Concept: Use the derivative formula: \[ \frac{d}{dx}[\cos^{-1}(u)] = \frac{-1}{\sqrt{1-u^2}} \cdot \frac{du}{dx} \] Also use the identity: \[ \frac{1 - x^2}{1 + x^2} = \cos(2\tan^{-1} x) \] This simplifies the expression significantly.
Step 1: Let \[ y = \cos^{-1} \left( \frac{1 - x^2}{1 + x^2} \right) \] Using identity: \[ \frac{1 - x^2}{1 + x^2} = \cos(2\tan^{-1} x) \] So, \[ y = \cos^{-1}[\cos(2\tan^{-1} x)] \]
Step 2: Simplify inverse cosine \[ y = 2\tan^{-1} x \]
Step 3: Differentiate \[ \frac{dy}{dx} = 2 \cdot \frac{1}{1 + x^2} \] Final Answer: \[ \boxed{\frac{2}{1 + x^2}} \] Explanation: Recognizing the trigonometric identity avoids complicated differentiation and makes the problem straightforward.