Is it possible to simplify $3\sin(z)\cos^2(z)-\sin^3(z)$? I have to write $\sin(3z)$ only with $\sin$'s and $\cos$'s. By using the additions-theorem I obtained $3\sin(z)\cos^2(z)-\sin^3(z)$ and I don't know If I can continue simplifying it. 

It's not completely clear what "simplify" means in this context -- very often, two equivalent trigonometric formulas can be equally "simple" -- but one thing you can do is replace $\cos^2(z)$ with $1-\sin^2(z)$.  Then you'll have
$$3\sin(z)(1-\sin^2(z))-\sin^3(z)$$
$$3\sin(z)-3\sin^3(z)-\sin^3(z)$$
$$3\sin(z)-4\sin^3(z)$$
