If A = 30°; show that :

sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

sin 3 A = 4 sin A sin (60° - A) sin (60° + A)

L.H.S. = sin 3A

= sin (3 x 30°)

= sin 90°

= 1

$\text{R.H.S.}= \text{4 sin A sin (60° - A) sin (60° + A)}\\[1em] = \text{4 sin 30°. sin (60° - 30°). sin (60° + 30°)}\\[1em] = \text{4 sin 30°. sin 30°. sin 90°}\\[1em] = 4 \times \dfrac{1}{2} \times \dfrac{1}{2} \times 1\\[1em] = 4 \times \dfrac{1}{4}\\[1em] = 1$

∴ L.H.S. = R.H.S.

Hence, sin 3 A = 4 sin A sin (60° - A) sin (60° + A).