( 4 ^ - 1 1 3 ) = - Vidyadip

MCQ

$\sin \left( {4{{\tan }^{ - 1}}\frac{1}{3}} \right) = $

A
$\frac{{12}}{{25}}$
✓
$\frac{{24}}{{25}}$
C
$\frac{1}{5}$
D
None of these

✓

Answer

Correct option: B.

$\frac{{24}}{{25}}$

b
(b) $\sin \left( {4{{\tan }^{ - 1}}\frac{1}{3}} \right) = \sin \left[ {2{{\tan }^{ - 1}}\left( {\frac{{2/3}}{{1 - (1/9)}}} \right)} \right]$

$ = \sin \left[ {2{{\tan }^{ - 1}}\frac{3}{4}} \right] = \sin {\sin ^{ - 1}}\left( {\frac{{2 \times (3/4)}}{{1 + (9/16)}}} \right)$

$ = \frac{3}{2} \times \frac{{16}}{{25}} = \frac{{24}}{{25}}$

$\left( {\because 2{{\tan }^{ - 1}}x = {{\sin }^{ - 1}}\frac{{2x}}{{1 + {x^2}}}} \right)$

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