Question

If \[ y = \sin^{-1} x, \] then \[ (1 - x^2)y_2 - xy_1 = 0. \]

• A. \( 0 \)
• B. \( 1 \)
• C. \( 2 \)
• D. \( 2y \)

Hint:

For inverse trigonometric functions, differentiate carefully and use algebraic simplifications.

Answer 1

The Correct Option is A

Step 1: Differentiating \[ y_1 = \frac{1}{\sqrt{1-x^2}}. \] Differentiating again: \[ y_2 = \frac{x}{(1-x^2)^{3/2}}. \] Step 2: Substituting into the given equation \[ (1 - x^2)y_2 - xy_1 = (1-x^2) \cdot \frac{x}{(1-x^2)^{3/2}} - x \cdot \frac{1}{\sqrt{1-x^2}}. \] Simplifying, both terms cancel: \[ 0. \]