Question

Just another cute problem,

Prove that: \( \large \tan \frac \pi {16} + 2 \tan \frac \pi {8} + 4 = \cot \frac{\pi}{16} \)

Enjoy!

Answer 1

It's so adorable it burns my eyes

Answer 2

\[\tan \frac{\pi}{4}=1\]\[\frac{2 \tan \frac{\pi}{8}}{1-\tan^2\frac{\pi}{8} }=1\]\[\frac{1-\tan^2\frac{\pi}{8} }{\tan \frac{\pi}{8} }=2\]\[\cot \frac{\pi}{8}-\tan\frac{\pi}{8}=2\]\[\frac{1-\tan^2\frac{\pi}{16} }{2\tan \frac{\pi}{16} }-\tan\frac{\pi}{8} =2 \]\[\cot \frac{\pi}{16}-\tan\frac{\pi}{16}-2\tan\frac{\pi}{8} =4 \]\[\cot \frac{\pi}{16}=\tan\frac{\pi}{16}+2\tan\frac{\pi}{8} +4 \]