Question

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

• A. 2
• B. 3
• C. 0
• D. 4

Hint:

When differentiating inverse trigonometric functions, remember the key identities such as \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \). This helps in simplifying the given expressions.

Answer 1

The Correct Option is D

Step 1: Simplify the given equation.
The given equation is:
\[ y = (\sin^{-1} x)^2 + (\cos^{-1} x)^2 \]
Using the identity \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), we get:
\[ y = \left( \frac{\pi}{2} \right)^2 = \frac{\pi^2}{4} \]

Step 2: Differentiate the equation.
Now, differentiate \( y \) with respect to \( x \):
\[ \frac{dy}{dx} = 0 \quad (\text{since } y = \frac{\pi^2}{4} \text{ is constant}) \]

Step 3: Second derivative.
Differentiate again to get the second derivative:
\[ \frac{d^2y}{dx^2} = 0 \]

Step 4: Substitute in the given expression.
Now, substitute the values in the given expression \( (1 - x^2) \frac{d^2y}{dx^2} - x \frac{dy}{dx} \): \[ (1 - x^2) \cdot 0 - x \cdot 0 = 0 \]

Step 5: Conclusion.
Hence, the value of the expression is 0. Therefore, the correct answer is:
\[ \boxed{4} \]