Question

If \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \), where \( 0<a, b<\frac{\pi}{2} \), then \( a - b \) is:

• A. \(\tan^{-1}(3)\)
• B. \(\tan^{-1}\left(\frac{3}{13}\right)\)
• C. \(\tan^{-1}(5)\)
• D. \(\tan^{-1}\left(\frac{9}{13}\right)\)
• E. \(\tan^{-1}\left(\frac{5}{13}\right)\)

Hint:

When subtracting angles whose tangent values are known, use the tangent subtraction formula to find the tangent of the resulting angle, and then use the inverse tangent to find the angle itself.

Answer 1

The Correct Option is D

The tangent of a difference identity states: \[ \tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b} \] Substitute \( a = \tan^{-1}\left(\frac{4}{3}\right) \) and \( b = \tan^{-1}\left(\frac{1}{3}\right) \): \[ \tan(a - b) = \frac{\frac{4}{3} - \frac{1}{3}}{1 + \left(\frac{4}{3} \cdot \frac{1}{3}\right)} \] Simplify: \[ = \frac{\frac{3}{3}}{1 + \frac{4}{9}} = \frac{1}{1 + \frac{4}{9}} = \frac{1}{\frac{13}{9}} = \frac{9}{13} \] Therefore, the angle difference \( a - b \) is: \[ a - b = \tan^{-1}\left(\frac{9}{13}\right) \]