Find the value of $\tan \left( {{{\tan }^{ - 1}}\frac{2}{11} + {{\tan }^{ - 1}}\frac{7}{{24}}} \right)$

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Find the value of $\tan \left( {{{\tan }^{ - 1}}\frac{2}{11} + {{\tan }^{ - 1}}\frac{7}{{24}}} \right)$
1. 1
2. $\rm 1\over2$
3. $\rm 1\over 3$
4. $\rm 2\over3$

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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:

$\rm \Rightarrow \tan \left( {{{\tan }^{ - 1}}\frac{2}{11} + {{\tan }^{ - 1}}\frac{7}{{24}}} \right) = \tan \left( \begin{array}{l} {\tan ^{ - 1}}\left( {\frac{{\frac{2}{11} + \frac{2}{{24}}}}{{1 - \frac{14}{{624}}}}} \right)\\ \end{array} \right)$

$\rm \Rightarrow \tan \left( \begin{array}{l} {\tan ^{ - 1}}\left( {\frac{{\frac{{48 \ + \ 77}}{{264}}}}{{\frac{{250}}{{264}}}}} \right)\\ \end{array} \right)$

$\rm \Rightarrow \tan \ ( tan^{-1} \ (\frac {125}{250}))$

$\rm \Rightarrow \tan \ ( tan^{-1} \ (\frac {1}{2}))$

$\rm \Rightarrow \frac{1}{2}$

${\tan ^{ - 1}}x - {\tan ^{ - y}} = {\tan ^{ - 1}}\left( {\frac{{x - y}}{{1 + xy}}} \right),xy > 1$