Prove that $sin x 2 x + 2 sin 4 x + s i n 6 x = 4 {cos}^{2} x . sin 4 x$

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Solution

L.H.S $= sin 2 x + 2 sin 4 x + sin 6 x$

$= sin 2 x + sin 6 x + 2 sin 4 x$

$= 2 sin \frac{2 x + 6 x}{2} cos \frac{6 x - 2 x}{2} + 2 sin 4 x$

$= 2 sin x cos 2 x + 2 sin 4 x$

$= 2 sin 4 x (cos 2 x + 1)$

$= 2 sin 4 x (2 {cos}^{2} x)$

$= 4 {cos}^{2} x . sin 4 x$

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