Question

If \( y^2 = 100\tan^{-1}x + 45\sec^{-1}x + 100\cot^{-1}x + 45\cosec^{-1}x \), then \( \frac{dy}{dx} \) is

• A. \( \frac{x^2-1}{x^2+1} \)
• B. \( \frac{x^2+1}{x^2-1} \)
• C. \( 1 \)
• D. \( 0 \)
• E. \( \frac{1}{x\sqrt{x^2-1}} \)

Hint:

Recognize inverse trigonometric sum identities to reduce expressions to constants quickly.

Answer 1

The Correct Option is D

Concept: Use identities: \[ \tan^{-1}x + \cot^{-1}x = \frac{\pi}{2} \] \[ \sec^{-1}x + \cosec^{-1}x = \frac{\pi}{2} \]

Step 1: Group terms
\[ y^2 = 100(\tan^{-1}x + \cot^{-1}x) + 45(\sec^{-1}x + \cosec^{-1}x) \]

Step 2: Apply identities
\[ y^2 = 100 \cdot \frac{\pi}{2} + 45 \cdot \frac{\pi}{2} \]

Step 3: Simplify
\[ y^2 = \frac{145\pi}{2} \]

Step 4: Differentiate
Since RHS is constant: \[ \frac{d}{dx}(y^2) = 0 \] \[ 2y \frac{dy}{dx} = 0 \]

Step 5: Solve
\[ \frac{dy}{dx} = 0 \]

Step 6: Final Answer
\[ \boxed{0} \]