Taylor Series of $\tan x$

I found a nice general formula for the Taylor series of $\tan x$:

$$\tan x = \sum_{n\,=\,1}^\infty \frac {(-1)^{n-1}2^{2n} (2^{2n}-1) B_{2n}} {(2n)!} x^{2n – 1} $$

where $B_n$ are the Bernoulli numbers and $|x| < \dfrac {\pi} 2$.

I’ve tried Googling for a proof but didn’t find anything. Hints would be appreciated too.

I am using the typical definition of the Bernoulli numbers:

$$\frac x {e^x – 1} = \sum_{n\,=\,0}^\infty \frac {B_n x^n} {n!}$$

Solutions Collecting From Web of "Taylor Series of $\tan x$"

$$\frac z{e^z-1}+\frac z 2=1+\sum_{n=2}^\infty\frac{B_n}{n!}z^n$$
Replace $z$ with $2iz$ to get
Now use following trigonometric formula

The only non zero $B_n$ with odd index is $B_1=-1/2$, So
which is valid for $|x|<2\pi$. Applying this with $x=it$, we get for $|t|<2\pi$:
\sum_{n=0}^\infty\frac{(-1)^nB_{2n}}{(2n)!}t^{2n}= \frac{t}{2}\cot\left(\frac{t}{2}\right)
Now note that
\cot(t)-2\cot(2t)=\frac{1-\cos^2t }{\sin t\cos t}=\tan t
so, for $|t|<\dfrac{\pi}{2}$ we have
t\tan t&=t\cot(t)-2t\cot(2t)\cr
From this the desired expansion follows.$\qquad\square$.