I can not prove the following with the properties of the trigonometric functions. Let
$$ p_{\pm}=\frac{pL}{\hbar}\pm n\pi. $$
I have the following expression:
$$ \Phi(p)=L\sqrt{\frac{2}{\pi L \hbar}}\frac{e^{-\frac{ip_{+}}{2}}}{p_{+}p_{-}} \left [ (-1)^{n}p_{+}\sin \left( \frac{p_{-}}{2}\right)- \sin\left( \frac{p_+}{2}\right) \right]. $$
I need to prove that:
$$ |\Phi(p)| ^{2}\propto\sin^2\left( \frac{p L}{\hbar}\right). $$