We know that

Replacing x with 3x and y with x, we get-

∴ sin 3x + sin x = 2 sin 2x cos x …(1)

Similarly,

We know that

Replacing x with 3x and y with x, we get-

∴ cos 3x - cos x = -2 sin 2x sin x …(2)

Now,

L.H.S = (sin 3x + sin x) sin x + (cos 3x – cos x) cos x

= (2 sin 2x cos x) sin x + (-2 sin 2x sin x) cos x

= 2 sin 2x cos x sin x - 2 sin 2x sin x cos x

= 0

= R.H.S

Hence, L.H.S = R.H.S…Proved