Prove that  $\cot \frac{\pi }{16}\cdot \mathrm{co}t\frac{2\pi }{16}\cdot \cot \frac{3\pi }{16}\dots \cot \frac{7\pi }{16}=1$

$\left(\cot \frac{\pi }{16}\cdot \cot \frac{7\pi }{16}\right)\cdot \left(\cot \frac{2\pi }{16}\cdot \cot \frac{6\pi }{16}\right)\cdot \left(\cot \frac{3\pi }{16}\cdot \cot \frac{5\pi }{16}\right)\cdot \cot \frac{4\pi }{16}$

$\left(\cot \frac{\pi }{16}\cdot \cot \left(\frac{\pi }{2}-\frac{\pi }{16}\right)\right)\left(\cot \frac{2\pi }{16}\cdot \cot \left(\frac{\pi }{2}-\frac{2\pi }{16}\right)\right)$$\left(\cot \frac{3\pi }{16}\cdot \cot \left(\frac{\pi }{2}-\frac{3\pi }{16}\right)\right)\cdot \cot \frac{4\pi }{16}$

$=\left(\cot \frac{\pi }{16}\cdot \tan \frac{\pi }{16}\right)\left(\cot \frac{2\pi }{16}\cdot \tan \frac{2\pi }{16}\right)\left(\cot \frac{3\pi }{16}\cdot \tan \frac{3\pi }{16}\right)\cdot \cot \frac{\pi }{4}$

$=1\times 1\times 1\times 1(\because \cot \theta \tan \theta =1)=1$