tan-¹2/11+tan-¹7/24=tan-¹1/2(prove)

To prove this identity, let's denote: - \( \theta_1 = \tan^{-1}\left(\frac{2}{11}\right) \) - \( \theta_2 = \tan^{-1}\left(\frac{7}{24}\right) \) We want to prove that: \[ \tan^{-1}\left(\frac{2}{11}\right) + \tan^{-1}\left(\frac{7}{24}\right) = \tan^{-1}\left(\frac{1}{2}\right) \] We'll use the tangent addition formula: \[ \tan(\alpha + \beta) = \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \cdot \tan \beta} \] Let's apply this formula: \[ \tan(\theta_1 + \theta_2) = \frac{\frac{2}{11} + \frac{7}{24}}{1 - \frac{2}{11} \cdot \frac{7}{24}} \] \[ = \frac{\frac{48 + 77}{264}}{1 - \frac{14}{264}} \] \[ = \frac{\frac{125}{264}}{\frac{250 - 14}{264}} \] \[ = \frac{\frac{125}{264}}{\frac{236}{264}} \] \[ = \frac{125}{236} \] Now, let's find \( \tan^{-1}\left(\frac{125}{236}\right) \): \[ \tan^{-1}\left(\frac{125}{236}\right) = \tan^{-1}\left(\frac{1}{2}\right) \] Therefore, we've proven the identity: \[ \tan^{-1}\left(\frac{2}{11}\right) + \tan^{-1}\left(\frac{7}{24}\right) = \tan^{-1}\left(\frac{1}{2}\right) \]