Question

4sin(π/3+x)sin(2π/3+x)=sin 3x
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Answer 1

Answer:

we know that

sin(A+B) × sin(A-B) = sin²A - sin²B.

apply this in LHS,

$4 \: sin( \frac{\pi}{3} + x) \: sin(\pi - \frac{\pi}{3} + x) = 4sin( \frac{\pi}{3} + x) \: sin( \frac{\pi}{3} - x) = 4( {sin}^{2} \frac{\pi}{3} - {sin}^{2} x) = 4( \frac{3}{4} - {sin}^{2} x) = 3 - 4 {sin}^{2} x = \frac{3sinx - 4 {sin}^{3} x}{sinx} = \frac{sin3x}{sinx} .$