We know that,

cos 3θ =4cos³θ–3cosθ

⇒4 cos³θ=cos3θ+3cosθ

And similarly

sin 3θ=3sin θ–4sin³ θ

⇒4 sin³θ=3sinθ–sin 3θ

Now,

Substituting the values from equation (i) and (ii), we get

(as sin(x+y) = sin x cos y+cos x sin y)

RHS

Hence Proved