Power Series Expansion for Tangent Function/Proof 2

Proof
We have:

Thus:

Replacing $x$ with $2 i x$ in the $(1)$:

Replacing $x$ with $2 i x$ in the $(1)$:

Then from Cotangent Minus Tangent:
 * $\tan x = \cot x - 2 \cot 2 x$

from which: