Given that ( - 2 )=2 ( + 2 ), then 2 2 is equal to - Vidyadip

MCQ

Given that $\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$, then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is equal to

A
$\frac{1}{2}$
✓
$\frac{1}{3}$
C
$\frac{1}{4}$
D
$\frac{1}{8}$

✓

Answer

Correct option: B.

$\frac{1}{3}$

(B)
$\cos \left(\frac{\alpha-\beta}{2}\right)=2 \cos \left(\frac{\alpha+\beta}{2}\right)$
$\Rightarrow \cos \frac{\alpha}{2} \cos \frac{\beta}{2}+\sin \frac{\alpha}{2} \sin \frac{\beta}{2}$
$=2 \cos \frac{\alpha}{2} \cos \frac{\beta}{2}-2 \sin \frac{\alpha}{2} \sin \frac{\beta}{2}$
$\Rightarrow 3 \sin \frac{\alpha}{2} \sin \frac{\beta}{2}=\cos \frac{\alpha}{2} \cos \frac{\beta}{2}$
$\Rightarrow \tan \frac{\alpha}{2} \tan \frac{\beta}{2}=\frac{1}{3}$

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