Question

Sin 4 theta hey, how do you simplify sin 4 theta? using double angle theory?

Answer 1

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