Correct Answer - Option 2 :
\(\rm 1\over2\)
Concept:
\({\tan ^{ - 1}}x + {\tan ^{ - y}} = {\tan ^{ - 1}}\left( {\frac{{x + y}}{{1 - xy}}} \right),xy > 1\)
Calculation:
Using the formula,
\({\tan ^{ - 1}}x + {\tan ^{ - y}} = {\tan ^{ - 1}}\left( {\frac{{x + y}}{{1 - xy}}} \right),xy > 1\)
\(\tan \left( {{{\tan }^{ - 1}}\frac{2}{11} + {{\tan }^{ - 1}}\frac{7}{{24}}} \right)\)
can be written as follows:
\(\rm \Rightarrow \tan \left( {{{\tan }^{ - 1}}\frac{2}{11} + {{\tan }^{ - 1}}\frac{7}{{24}}} \right) = \tan \left( \begin{array}{l} {\tan ^{ - 1}}\left( {\frac{{\frac{2}{11} + \frac{2}{{24}}}}{{1 - \frac{14}{{624}}}}} \right)\\ \end{array} \right)\)
\(\rm \Rightarrow \tan \left( \begin{array}{l} {\tan ^{ - 1}}\left( {\frac{{\frac{{48 \ + \ 77}}{{264}}}}{{\frac{{250}}{{264}}}}} \right)\\ \end{array} \right)\)
\(\rm \Rightarrow \tan \ ( tan^{-1} \ (\frac {125}{250}))\)
\(\rm \Rightarrow \tan \ ( tan^{-1} \ (\frac {1}{2}))\)
\( \rm \Rightarrow \frac{1}{2}\)
\({\tan ^{ - 1}}x - {\tan ^{ - y}} = {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right),xy > 1\)