Q13/054) prove that:sin5x-2sinx(cos2x+cos4x) = sin x

we have 2 sin A cos B = sin ( A + B ) + sin (A- B) ..1

Putting A = x and B = 2x we get

2 sin x cos 2x = sin 3x – sin x ..2

Putting A = x and B = 4x we get

2 sin x cos 4x = sin 5x – sin 3x ..3

Add (2) and (3) to get 2 sin x cos 2x + 2 sin x cos 4x = sin 5x – sin x

Or sin 5x – (2 sin x cos 2x + 2 sin x cos 4x)= sin x

Or sin 5x – 2 sin x (cos 2x + cos 4x)= sin x

Proved