$$\begin{equation}\begin{split} \therefore\,\tan\,w= \frac{\sin\,{\it 2}\,w}{\cos\,{\it 2}\,w+ {\it 1}} \end{split}\end{equation}$$
$$\begin{equation}\begin{split} \therefore\,\tan\,\frac{x+ w}{{\it 2}}= \frac{\sin\,x+ \sin\,w}{\cos\,x+ \cos\,w} \end{split}\end{equation}$$
$$\begin{equation}\begin{split} \therefore\,{\it 2}\sin(\,2\,\pi\,w\,)\,/\,{\it 2}\sin(\,{\it 2}\,\pi(\,w+ {\it 1}/{\it 2}\,)\,)= -\,{\it 1} \end{split}\end{equation}$$