\(\tan A+2 \tan 2 A+4 \tan 4 A+8 \operatorname{Cot} 8 A=\)

\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+8 \times \frac{1-\tan ^{2} 4 \mathrm{~A}}{2 \tan 4 \mathrm{~A}}\)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \tan 4 \mathrm{~A}+\frac{4\left(1-\tan ^{2} 4 \mathrm{~A}\right)}{\tan 4 \mathrm{~A}}\)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{4 \tan ^{2} 4 \mathrm{~A}+4-4 \tan ^{2} 4 \mathrm{~A}}{\tan 4 \mathrm{~A}}\)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{4}{\tan 4 \mathrm{~A}}=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \cot 4 \mathrm{~A}\)
\(=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+4 \times \frac{1-\tan ^{2} 2 \mathrm{~A}}{2 \tan 2 \mathrm{~A}}=\tan \mathrm{A}+2 \tan 2 \mathrm{~A}+\frac{2\left(1-\tan ^{2} 2 \mathrm{~A}\right)}{\tan 2 \mathrm{~A}}\)
\(=\tan \mathrm{A}+\frac{2 \tan ^{2} 2 \mathrm{~A}+2-2 \tan ^{2} 2 \mathrm{~A}}{\tan 2 \mathrm{~A}}=\tan \mathrm{A}+\frac{2}{\tan 2 \mathrm{~A}}\)
\(=\tan \mathrm{A}+2 \cot 2 \mathrm{~A}=\frac{2\left(1-\tan ^{2} \mathrm{~A}\right)}{2 \tan \mathrm{A}}=\tan \mathrm{A}+\frac{1-\tan ^{2} \mathrm{~A}}{\tan \mathrm{A}}\)
\(=\frac{\tan ^{2} \mathrm{~A}+1-\tan ^{2} \mathrm{~A}}{\tan \mathrm{A}}=\frac{1}{\tan \mathrm{A}}=\cot \mathrm{A}\)