Yes, you do use double-angle formulas.
You begin with double-angle formula for sine:
sin(4θ) = sin(2·2θ) = 2·sin(2θ)·cos(2θ)
Then, double-angle formula for sine again:
2·sin(2θ)·cos(2θ) = 2·2·sinθ·cosθ·cos(2θ) = 4·sinθ·cosθ·cos(2θ)
Then double-angle formula for cosine:
4·sinθ·cosθ·cos(2θ) = 4·sinθ·cosθ·(cos²θ - sin²θ)
Last, multiply:
4·sinθ·cosθ·(cos²θ - sin²θ) = 4·sinθ·cos³θ - 4·sin³θ·cosθ
