We know that-

Now,

L.H.S = sin x + sin 3x + sin 5x + sin 7x

= (sin x + sin 7x) + (sin 3x + sin 5x)

= 2 sin 4x cos 3x + 2 sin 4x cos x

= 2 sin4x (cos 3x + cos x)

= 2 sin 4x (2 cos 2x cos x)

= 4 cos x cos 2x sin 4x

= R.H.S

Hence, L.H.S = R.H.S…Proved