If 2 A = 3 B, then 2B 5 - 2B is equal to - Vidyadip

MCQ

If $2\tan A = 3\tan B,$ then $\frac{{\sin 2B}}{{5 - \cos 2B}}$ is equal to

A
$\tan A - \tan B$
✓
$\tan (A - B)$
C
$\tan (A + B)$
D
$\tan (A + 2B)$

✓

Answer

Correct option: B.

$\tan (A - B)$

b
(b) $2\tan {\rm A} = 3\tan B$

==> $\tan A = \frac{3}{2}\tan B = \frac{3}{2}t$, [Let $\tan B = t$]

==> $\sin 2B = \frac{{2t}}{{1 + {t^2}}},\cos 2B = \frac{{1 - {t^2}}}{{1 + {t^2}}}$

$\therefore$ $\frac{{\left( {\frac{{2t}}{{1 + {t^2}}}} \right)}}{{5 - \left( {\frac{{1 - {t^2}}}{{1 + {t^2}}}} \right)}}$

$ = \frac{{2t}}{{4 + 6{t^2}}} = \frac{t}{{2 + 3{t^2}}} = \tan (A - B)$.

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