Let $P\left(k\right)=(1+\cos \frac{\pi }{4k})(1+\cos \frac{(2k-1)\pi }{4k})(1+\cos \frac{(2k+1)\pi }{4k})(1+\cos \frac{(4k-1)\pi }{4k})$ then 

$P\left(k\right)=(1+\cos \frac{\pi }{4k})(1+\cos \frac{(2k-1)\pi }{4k})(1+\cos \frac{(2k+1)\pi }{4k})(1+\cos \frac{(4k-1)\pi }{4k})$

$=(1+\cos \frac{\pi }{4k})(1+\sin \frac{\pi }{4k})(1-\sin \frac{\pi }{4k})(1-\cos \frac{\pi }{4k})$

$=(1-{\cos }^2\frac{\pi }{4k})(1-{\sin }^2\frac{\pi }{4k})$

$={\sin }^2\frac{\pi }{4k}\cdot {\cos }^2\frac{\pi }{4k}$

$P\left(k\right)=\frac{1}{4}{\sin }^2\left(\frac{\pi }{2k}\right)$

$P\left(3\right)=\frac{1}{4}\cdot \frac{1}{4}=\frac{1}{16}$

$P\left(4\right)=\frac{1}{4}{\sin }^2\frac{\pi }{8}=\frac{1}{8}(1-\cos \frac{\pi }{4})=\frac{2-\sqrt{2}}{16}$

$P\left(5\right)=\frac{1}{4}{\sin }^2\frac{\pi }{10}=\frac{1}{8}(1-\cos 36°)=\frac{3-\sqrt{5}}{32}$

$P\left(6\right)=\frac{1}{4}{\sin }^2\frac{\pi }{12}=\frac{1}{8}(1-\cos \frac{\pi }{6})=\frac{2-\sqrt{3}}{16}$.