Official solution
We will use here those two formulas,
sin2 \[\theta \] = \[{{1 - \cos 2\theta } \over 2}\] and cos2 \[\theta \] = \[{{1 + \cos 2\theta } \over 2}\]
L = sin2\[\left( {{\pi \over {16}}} \right)\] - sin2\[\left( {{\pi \over {8}}} \right)\]
\[ \Rightarrow \] L = \[\left( {{{1 - \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)\] - \[\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)\]
\[ \Rightarrow \] L = \[{1 \over 2}\left( {\cos \left( {{\pi \over 4}} \right) - \cos \left( {{\pi \over 8}} \right)} \right)\]
\[ \Rightarrow \] L = \[{1 \over {2\sqrt 2 }} - {1 \over 2}\cos \left( {{\pi \over 8}} \right)\]
M = cos2\[\left( {{\pi \over {16}}} \right)\] - sin2\[\left( {{\pi \over {8}}} \right)\]
\[ \Rightarrow \] M = \[\left( {{{1 + \cos \left( {{\pi \over 8}} \right)} \over 2}} \right)\] - \[\left( {{{1 - \cos \left( {{\pi \over 4}} \right)} \over 2}} \right)\]
\[ \Rightarrow \] M = \[{1 \over {2\sqrt 2 }} + {1 \over 2}\cos {\pi \over 8}\]