\( \tan (\tan^{-1} (\frac{1}{3}) + \tan^{-1} (\frac{1}{2})) = ? \)

Another Explanation (5): প্রশ্ন: \( \tan \left( \tan^{-1} \left(\frac{1}{3}\right) + \tan^{-1} \left(\frac{1}{2}\right) \right) = ? \) উত্তর: "1" সমাধান: প্রথমে, ধরা যাক: \[ A = \tan^{-1} \left(\frac{1}{3}\right), \quad B = \tan^{-1} \left(\frac{1}{2}\right) \] অর্থাৎ, \[ \tan A = \frac{1}{3}, \quad \tan B = \frac{1}{2} \] আমরা জানি, \[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \] অতএব, \[ \tan (A + B) = \frac{\frac{1}{3} + \frac{1}{2}}{1 - \left(\frac{1}{3} \times \frac{1}{2}\right)} \] সাধারণ করে, \[ = \frac{\frac{2}{6} + \frac{3}{6}}{1 - \frac{1}{6}} = \frac{\frac{5}{6}}{\frac{5}{6}} = 1 \] অতএব, \[ \tan \left( \tan^{-1} \left(\frac{1}{3}\right) + \tan^{-1} \left(\frac{1}{2}\right) \right) = \boxed{1} \]