If 3 =4 (x+ ) (x- ), then x= - Vidyadip

MCQ

If $\sin 3 \alpha=4 \sin \alpha \sin (x+\alpha) \sin (x-\alpha)$, then $x=$

A
$n \pi \pm \frac{\pi}{6}$
✓
$n \pi \pm \frac{\pi}{3}$
C
$n \pi \pm \frac{\pi}{4}$
D
$n \pi \pm \frac{\pi}{2}$

✓

Answer

Correct option: B.

$n \pi \pm \frac{\pi}{3}$

(B) $\sin 3 \alpha=4 \sin \alpha \sin (x+\alpha) \sin (x-\alpha)$
$\therefore \sin 3 \alpha=4 \sin \alpha\left(\sin ^2 x \cos ^2 \alpha-\cos ^2 x \sin ^2 \alpha\right)$
$\therefore 3 \sin \alpha-4 \sin ^3 \alpha=4 \sin \alpha\left(\sin ^2 x-\sin ^2 \alpha\right)$
$\therefore \quad \sin ^2 x=\left(\frac{3}{4}\right) \Rightarrow \sin ^2 x=\sin ^2 \frac{\pi}{3}$
$\therefore \quad x= n \pi \pm \frac{\pi}{3}$

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