$\cos \frac{\pi }{4}\cos \frac{\pi }{8}\cos \frac{\pi }{16}\cos \frac{\pi }{32}=2^{m}cosec\frac{\pi }{n}$

$LHS=\frac{1}{2\sin \left({\displaystyle \frac{\pi }{2^n}}\right)}\left[2\cos \left(\frac{\pi }{2^n}\right)\sin \left(\frac{\pi }{2^n}\right)\right]\cos \frac{\pi }{2^{n-1}}..........\cos \frac{\pi }{4}$

$\frac{1}{2}\cos \left(\frac{\pi }{2^n}\right)\left[2\sin \left(\frac{\pi }{2^{n-1}}\right)\cos \left(\frac{\pi }{2^{n-1}}\right)\right]\cos \frac{\pi }{2^{n-2}}..........\cos \frac{\pi }{4}$

$\frac{1}{2\times 2^2}\cos ec\left(\frac{\pi }{2^n}\right)\left[2\sin \left(\frac{\pi }{2^{n-2}}\right)\cos \left(\frac{\pi }{2^{n-2}}\right)\right]\cos \frac{\pi }{2^{n-2}}..........\cos \frac{\pi }{4}$

$\frac{1}{2^{n-2}}\cos ec\left(\frac{\pi }{2^n}\right)\left[2\sin \left(\frac{\pi }{8}\right)\cos \left(\frac{\pi }{8}\right)\right]\cos \frac{\pi }{4}$

$\frac{1}{2\times 2^{n-2}}\cos ec\left(\frac{\pi }{2^n}\right)\left[2\sin \left(\frac{\pi }{4}\right)\cos \left(\frac{\pi }{4}\right)\right]$

$\frac{1}{2^{n-1}}\cos ec\left(\frac{\pi }{2^n}\right)\left[\sin \left(\frac{\pi }{2}\right)\right]$

$\frac{1}{2^{n-1}}\cos ec\left(\frac{\pi }{2^n}\right)$

if n=5 m=23,   m+n=28