We have,
L.H.S.= sinx + sin3x + sin5x + sin7x = (sinx + sin 7x) + (sin 3x + sin 5x)
= 2 sin (

x+7x

2

) cos (

x−7x

2

) + 2 sin (

3x+5x

2

) cos (

3x−5x

2

)
= 2 sin (4x) cos (–3x) + 2 sin 4x cos (–x)
= 2 sin 4x cos 3x + 2 sin 4x cos x [Since cos (–θ) = cos θ]
= 2 sin 4x(cos 3x + cosx)
= 2 sin 4x (2cos(

3x+x

2

)cos(

3x−x

2

))
= 2 sin 4x(2cos 2x cosx) = 4 cos x cos 2x sin 4x = R.H.S.