Relations & Functions: Inverse Trigonometric Identities

Using $\tan^{-1}x + \tan^{-1}y = \tan^{-1}\left(\frac{x + y}{1 - xy}\right)$ since $xy = \frac{1}{10} < 1$: $\tan^{-1}\left(\frac{1}{2}\right) + \tan^{-1}\left(\frac{1}{5}\right) = \tan^{-1}\left(\frac{\frac{1}{2} + \frac{1}{5}}{1 - \frac{1}{10}}\right) = \tan^{-1}\left(\frac{\frac{7}{10}}{\frac{9}{10}}\right) = \tan^{-1}\left(\frac{7}{9}\right)$

Now add $\tan^{-1}\left(\frac{1}{8}\right)$ with $xy = \frac{7}{9} \times \frac{1}{8} = \frac{7}{72} < 1$: $\tan^{-1}\left(\frac{7}{9}\right) + \tan^{-1}\left(\frac{1}{8}\right) = \tan^{-1}\left(\frac{\frac{7}{9} + \frac{1}{8}}{1 - \frac{7}{72}}\right) = \tan^{-1}\left(\frac{\frac{56 + 9}{72}}{\frac{72 - 7}{72}}\right) = \tan^{-1}\left(\frac{65}{65}\right) = \tan^{-1}(1)$

$\tan^{-1}(1) = \frac{\pi}{4} = \text{RHS}$