The value of $$3\,{\tan ^{ - 1}}\frac{1}{2} + 2\,{\tan ^{ - 1}}\frac{1}{5} + {\sin ^{ - 1}}\frac{{142}}{{65\sqrt 5 }}{\text{ is}}$$

Solution :
$$\eqalign{ & {\text{We have,}} \cr & 3\,{\tan ^{ - 1}}\frac{1}{2} + 2\,{\tan ^{ - 1}}\frac{1}{5} + {\sin ^{ - 1}}\frac{{142}}{{65\sqrt 5 }} \cr & = 2\left( {{{\tan }^{ - 1}}\frac{1}{2} + {{\tan }^{ - 1}}\frac{1}{5}} \right) + {\tan ^{ - 1}}\frac{1}{2} + {\tan ^{ - 1}}\frac{{142}}{{31}} \cr & = 2\,{\tan ^{ - 1}}\frac{7}{9} + \pi + {\tan ^{ - 1}}\frac{{\frac{1}{2} + \frac{{142}}{{31}}}}{{1 - \left( {\frac{1}{2}} \right)\left( {\frac{{142}}{{31}}} \right)}} \cr & = {\tan ^{ - 1}}\frac{{\frac{{14}}{9}}}{{1 - \frac{{49}}{{81}}}} + \pi - {\tan ^{ - 1}}\frac{{315}}{{80}} \cr & = \pi - {\tan ^{ - 1}}\frac{{63}}{{16}} + {\tan ^{ - 1}}\frac{{63}}{{16}} = \pi \cr} $$