Could I get anymore Tanner?

Geometry Level 3

\[\tan(\alpha)+2\tan(2\alpha)+4\tan(4\alpha)+8\cot(8\alpha)=\cot(n\alpha)\]

If the above trigonomtric equation is true for all \(\alpha\), what is the value of \(n\)?

This problem is part of the set Trigonometry.

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