Underwood Dudley trigonometry formula for a trisection

Question

In his publications on trisectors Dudley claims that one construction is equivalent to the following formula: $$\sin\left(\frac{\alpha}{3}\right)=\frac{\sin \alpha}{2+\cos \alpha}$$ I was able to derive the triple angle formulas in Wikipedia, but when applying them to the above formula, I always end up with $\cos^2\alpha$ or $\cos 2\alpha$ instead of $\cos \alpha$.