If $L={\sin }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)$ and $M={\cos }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right),$ then

$\begin{array}{l}L={\sin }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)\\ =\frac{1-\cos (\pi /8)}{2}-\left(\frac{1-\cos (\pi /4)}{2}\right)\\ =\frac{1}{2}[\cos \left(\frac{\pi }{4}\right)-\cos \left(\frac{\pi }{8}\right)]\\ L=\frac{1}{2\sqrt{2}}-\frac{1}{2}\cos \left(\frac{\pi }{8}\right)\\ M={\cos }^2\left(\frac{\pi }{16}\right)-{\sin }^2\left(\frac{\pi }{8}\right)\\ =\frac{1+\cos (\pi /8)}{2}-\left(\frac{1-\cos (\pi /4)}{2}\right)\end{array}$

$\begin{array}{l}\\ =\frac{1}{2}(\cos (\pi /8)+\cos \left(\frac{\pi }{4}\right))\\ M=\frac{1}{2}\cos \left(\frac{\pi }{8}\right)+\frac{1}{2\sqrt{2}}\end{array}$