$\tan \mathrm{\theta }+2 \tan 2\mathrm{\theta }+4 \tan 4\mathrm{\theta }+8 \tan 8\mathrm{\theta }+16 \tan 16\mathrm{\theta }+32 \cot 32\mathrm{\theta }=$

$$\begin{gathered} cot\theta -\tan \theta =2cot2\theta \\ \tan \mathrm{\theta }+2 \tan 2\mathrm{\theta }+4 \tan 4\mathrm{\theta }+8 \tan 8\mathrm{\theta }+16 \tan 16\mathrm{\theta }+32 \cot 32\mathrm{\theta } \\ =\mathrm{cot\theta }-2\cot 2\mathrm{\theta }+2(\cot 2\mathrm{\theta }-2\cot 4\mathrm{\theta })+4(\cot 4\mathrm{\theta }-2\cot 8\mathrm{\theta })+8(\cot 8\mathrm{\theta }-2\cot 16\mathrm{\theta }) \\ +16(\cot 16\mathrm{\theta }-2\cot 32\mathrm{\theta })+32\cot 32\mathrm{\theta } \\ =\mathrm{cot\theta } \end{gathered}$$