Question

tan\(^{-1} 2 - \) tan\(^{-1} \frac{1}{3}\) is equal to:

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{3} \)
• C. \( \frac{\pi}{4} \)
• D. \( \frac{\pi}{6} \)
• E. 0

Hint:

Use the identity \( \tan^{-1} a - \tan^{-1} b = \tan^{-1} \left( \frac{a-b}{1 + ab} \right) \) to simplify differences of inverse tangents.

Answer 1

The Correct Option is C

We are given the expression: \[ {tan}^{-1} 2 - {tan}^{-1} \left( \frac{1}{3} \right). \] To solve this, we use the following formula for the difference of two inverse tangents: \[ {tan}^{-1} a - {tan}^{-1} b = {tan}^{-1} \left( \frac{a - b}{1 + ab} \right), \] where \( a = 2 \) and \( b = \frac{1}{3} \). 
Substitute these values into the formula: \[ {tan}^{-1} 2 - {tan}^{-1} \left( \frac{1}{3} \right) = {tan}^{-1} \left( \frac{2 - \frac{1}{3}}{1 + 2 \cdot \frac{1}{3}} \right). \] Now simplify the numerator and denominator: \[ \frac{2 - \frac{1}{3}}{1 + \frac{2}{3}} = \frac{\frac{6}{3} - \frac{1}{3}}{\frac{3}{3} + \frac{2}{3}} = \frac{\frac{5}{3}}{\frac{5}{3}} = 1. \] Thus, we have: \[ {tan}^{-1} 2 - {tan}^{-1} \left( \frac{1}{3} \right) = {tan}^{-1} 1. \] Since \( {tan}^{-1} 1 = \frac{\pi}{4} \), we conclude that: \[ {tan}^{-1} 2 - {tan}^{-1} \left( \frac{1}{3} \right) = \frac{\pi}{4}. \] Thus, the correct answer is \( \boxed{\frac{\pi}{4}} \), corresponding to option (C)