Sin 4x (1 + 2 cos x) = sin 3x + sin 4x + sin 5x

Trigonometry Identity.

sin A + sin B = 2 sin 1/2 (A + B) cos 1/2 (A - B)
cos (-θ) = cos θ

From right side:
(sin 3x + sin 5x) + sin 4x
= [2 sin 1/2 (3x + 5x) cos 1/2 (3x - 5x)] + sin 4x
= (2 sin 4x cos x) + sin 4x
= sin 4x (2 cos x + 1)
Proven, sin 4x (1 + 2 cos x) = sin 3x + sin 4x + sin 5x