${\tan ^{ - 1}}\frac{2}{{11}} + {\tan ^{ - 1}}\frac{7}{{24}} = {\tan ^{ - 1}}\frac{1}{2}$

${\tan ^{ - 1}}\left( {\frac{2}{{11}}} \right) + {\tan ^{ - 1}}\left( {\frac{7}{{24}}} \right) = {\tan ^{ - 1}}\left[ {\frac{{\frac{2}{{11}} + \frac{7}{{24}}}}{{1 - \frac{2}{{11}} \times \frac{7}{{24}}}}} \right]$

$= {\tan ^{ - 1}}\left[ {\frac{{\frac{{18 + 77}}{{264}}}}{{\frac{{264 - 14}}{{264}}}}} \right] = {\tan ^{ - 1}}\left[ {\frac{{125}}{{250}}} \right] = {\tan ^{ - 1}}\left( {\frac{1}{2}} \right)$