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

MCQ

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

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

✓

Answer

Correct option: B.

$\tan (A-B)$

(B)
$2 \tan A=3 \tan B$
$\Rightarrow \tan A =\frac{3}{2} \tan B=\frac{3}{2} t \quad[$ Let $\tan B =t]$
$\sin 2 B=\frac{2 t }{1+ t ^2}, \cos 2 B=\frac{1- t ^2}{1+ t ^2}$
$\therefore \frac{\sin 2 B}{5-\cos 2 B}=\frac{\left(\frac{2 t}{1+t^2}\right)}{5-\left(\frac{1-t^2}{1+t^2}\right)}=\frac{2 t}{4+6 t^2}$
$=\frac{t}{2+3 t^2}=\frac{\frac{3 t}{2}-t}{1+\frac{3 t^2}{2}}$
$=\tan ( A - B )$

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