$\begin{array}{l}
M = {\cos ^2}\left( {\dfrac{\pi }{4} + \alpha } \right) - {\cos ^2}\left( {\dfrac{\pi }{4} - \alpha } \right)\\
M = \left[ {\cos \left( {\dfrac{\pi }{4} + \alpha } \right) - \cos \left( {\dfrac{\pi }{4} - \alpha } \right)} \right]\left[ {\cos \left( {\dfrac{\pi }{4} + \alpha } \right) + \cos \left( {\dfrac{\pi }{4} - \alpha } \right)} \right]\\
M = - 2\sin \dfrac{\pi }{4}\sin \alpha .2.\cos \dfrac{\pi }{4}\cos \alpha \\
M = - \sin \dfrac{\pi }{2}\sin 2\alpha = - \sin 2\alpha
\end{array}$
