^ - 1 ( 1 4 ) + ^ - 1 ( 2 9 ) = - Vidyadip

MCQ

${\tan ^{ - 1}}\left( {\frac{1}{4}} \right) + {\tan ^{ - 1}}\left( {\frac{2}{9}} \right) = $

✓
$\frac{1}{2}{\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$
B
$\frac{1}{2}{\sin ^{ - 1}}\left( {\frac{3}{5}} \right)$
C
$\frac{1}{2}{\tan ^{ - 1}}\left( {\frac{3}{5}} \right)$
D
None of these

✓

Answer

Correct option: A.

$\frac{1}{2}{\cos ^{ - 1}}\left( {\frac{3}{5}} \right)$

a
(a) ${\tan ^{ - 1}}\frac{1}{4} + {\tan ^{ - 1}}\frac{2}{9} = {\tan ^{ - 1}}\left( {\frac{{(1/4) + (2/9)}}{{1 - (1/4) \times (2/9)}}} \right)$

$={\tan ^{ - 1}}\left( {\frac{1}{2}} \right) = \frac{1}{2}.2{\tan ^{ - 1}}\left( {\frac{1}{2}} \right) = \frac{1}{2}{\tan ^{ - 1}}\frac{{2(1/2)}}{{1 - (1/4)}}$

$ = \frac{1}{2}{\tan ^{ - 1}}\frac{4}{3} = \frac{1}{2}{\cos ^{ - 1}}\frac{3}{5}$.

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