Question

Prove that : sin 2x + 2sin 4x + sin 6x = 4cos^2 x sin 4x

Answer 1

Please see the attachment

Answer 2

$sin6x + sin2x + 2sin4x \\ \\ = 2sin( \frac{6x + 2x}{2} )cos( \frac{6x - 2x}{2} ) + 2sin4x \\ \\ = 2sin4xcos2x + 2sin4x \\ \\ = 2sin4x(cos2x + 1) \\ \\ = 2sin4x(2 {cos}^{2} x - 1 + 1) \\ \\ = 2sin4x(2 {cos}^{2} x) \\ \\ = 4 {cos}^{2} xsin4x$