Question

If \( 3 \tan^{-1}(x) + \cot^{-1}(x) = \pi \), then \( \sin^{-1}(x) \) is:

• A. \( \frac{\pi}{12} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)
• E. \( \frac{\pi}{2} \)

Hint:

Use the identity for inverse trigonometric functions to simplify the equation and solve for \( x \).

Answer 1

Answer: (E)

Step 1: We are given \( 3 \tan^{-1}(x) + \cot^{-1}(x) = \pi \). 
Using the identity \( \cot^{-1}(x) = \frac{\pi}{2} - \tan^{-1}(x) \), we get: \[ 3 \tan^{-1}(x) + \left( \frac{\pi}{2} - \tan^{-1}(x) \right) = \pi \] Simplifying: \[ 2 \tan^{-1}(x) = \frac{\pi}{2} \] \[ \tan^{-1}(x) = \frac{\pi}{4} \] Thus, \( x = 1 \). 
Step 2: Now, calculate \( \sin^{-1}(x) \) for \( x = 1 \): \[ \sin^{-1}(1) = \frac{\pi}{2} \]