Find the value of $4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}$.

We have, $4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}$

$= 2 \cdot 2{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}$

$= 2 \cdot \left[ {{{\tan }^{ - 1}}\frac{{\frac{2}{5}}}{{1 - {{\left( {\frac{1}{5}} \right)}^2}}}} \right] - {\tan ^{ - 1}}\frac{1}{{239}}$

$= 2 \cdot \left[ {{{\tan }^{ - 1}}\left( {\frac{{\frac{2}{5}}}{{1 - \frac{1}{{25}}}}} \right)} \right] - {\tan ^{ - 1}}\frac{1}{{239}}$

$= 2 \cdot \left[ {{{\tan }^{ - 1}}\left( {\frac{{2/5}}{{24/25}}} \right)} \right] - {\tan ^{ - 1}}\frac{1}{{239}}$

$= 2{\tan ^{ - 1}}\frac{5}{{12}} - {\tan ^{ - 1}}\frac{1}{{239}}$

$= {\tan ^{ - 1}}\frac{{2 \cdot \frac{5}{{12}}}}{{1 - {{\left( {\frac{5}{{12}}} \right)}^2}}} - {\tan ^{ - 1}}\frac{1}{{239}}$

$= {\tan ^{ - 1}}\left( {\frac{{\frac{5}{6}}}{{1 - \frac{{25}}{{144}}}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}$

$= {\tan ^{ - 1}}\left( {\frac{{144 \times 5}}{{119 \times 6}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}$
$= {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) - {\tan ^{ - 1}}\frac{1}{{239}}$

$= {\tan ^{ - 1}}\left( {\frac{{\frac{{120}}{{119}} - \frac{1}{{239}}}}{{1 + \frac{{120}}{{119}} \cdot \frac{1}{{239}}}}} \right)$

$= {\tan ^{ - 1}}\left( {\frac{{120 \times 239 - 119}}{{119 \times 239 + 120}}} \right)$

$= {\tan ^{ - 1}}\left[ {\frac{{28680 - 119}}{{28441 + 120}}} \right] = {\tan ^{ - 1}}\frac{{28561}}{{28561}}$

$= {\tan ^{ - 1}}(1) = {\tan ^{ - 1}}\left( {\tan \frac{\pi }{4}} \right) = \frac{\pi }{4}$