4 ^ - 1 1 5 - ^ - 1 1 239 is equal to - Vidyadip

MCQ

$4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}}$ is equal to

A
$\pi $
B
$\frac{\pi }{2}$
C
$\frac{\pi }{3}$
✓
$\frac{\pi }{4}$

✓

Answer

Correct option: D.

$\frac{\pi }{4}$

d
(d) Since $2{\tan ^{ - 1}}x = {\tan ^{ - 1}}\frac{{2x}}{{1 - {x^2}}}$

$\therefore$ $4{\tan ^{ - 1}}\frac{1}{5} = 2\,\left[ {2{{\tan }^{ - 1}}\frac{1}{5}} \right] = 2{\tan ^{ - 1}}\frac{{\frac{2}{5}}}{{1 - \frac{1}{{25}}}}$

$ = 2{\tan ^{ - 1}}\frac{{10}}{{24}} = {\tan ^{ - 1}}\frac{{\frac{{20}}{{24}}}}{{1 - \frac{{100}}{{576}}}} = {\tan ^{ - 1}}\frac{{120}}{{119}}$

So, $4{\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{239}} = {\tan ^{ - 1}}\frac{{120}}{{119}} - {\tan ^{ - 1}}\frac{1}{{239}}$

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

==> ${\tan ^{ - 1}}1 = \frac{\pi }{4}$.

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