$\cot \frac{\pi }{16}· \cot \frac{2\pi }{16}· \cot \frac{3\pi }{16}· \cot \frac{4\pi }{16}· \cot \frac{5\pi }{16}· \cot \frac{6\pi }{16}· \cot \frac{7\pi }{16}=$

$\cot \frac{\pi }{16}· \cot \frac{2\pi }{16}· \cot \frac{3\pi }{16}· \cot \frac{4\pi }{16}· \cot \frac{5\pi }{16}· \cot \frac{6\pi }{16}· \cot \frac{7\pi }{16}$

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

$=\cot \frac{\pi }{16}· \cot \frac{2\pi }{16}· \cot \frac{3\pi }{16}· \cot \frac{4\pi }{16}· \tan \frac{5\pi }{16}·\tan \frac{6\pi }{16}· \tan \frac{7\pi }{16}$

$=\bcancel{\cot \frac{\pi }{16}}· \bcancel{\cot \frac{2\pi }{16}}· \bcancel{\cot \frac{3\pi }{16}}· \cot \frac{4\pi }{16}· \bcancel{\tan \frac{5\pi }{16}}·\bcancel{\tan \frac{6\pi }{16}}· \bcancel{\tan \frac{7\pi }{16}}$

$=\cot \left(\frac{\pi }{4}\right)$

$=1$