<s>[INST] Continued fraction for $\tan(nx)$ I found this beautiful continued fraction expansion of $\tan(nx)$, $n$ being a positive integer, online but I don't remember the source now:
$\displaystyle \tan(nx) = \cfrac{n\tan x}{1 -\cfrac{(n^{2} - 1^{2})\tan^{2}x}{3 -\cfrac{(n^{2} - 2^{2})\tan^{2}x}{5 -\cfrac{(n^{2} - 3^{2})\tan^{2}x}{7 -\cdots}}}}$
the last term in the continued fraction being $\dfrac{(n^{2} - (n - 1)^{2})\tan^{2}x}{(2n - 1)} = \tan^{2}x$. For example for $n = 3$ we have
$\displaystyle \tan 3x = \cfrac{3\tan x}{1 -\cfrac{8\tan^{2}x}{3-\cfrac{5\tan^{2}x}{5}}}$
which is correct and can be verified easily. I would like to know a simple proof via elementary trigonometry and algebra.
I had asked this question long back on NRICH(https://nrich.maths.org/discus/messages/153904/145455.html) but did not get any helpful answer. The reason I wish to know the proof is that it will lead me to an elementary proof of continued fraction expansion of $\tan z$. Putting $nx = z$ where $z$ is kept constant and $n$ is made to tend to $\infty$ (so that $x \to 0$) we get
$\displaystyle \tan z = \cfrac{z}{1-\cfrac{z^{2}}{3 -\cfrac{z^{2}}{5-\cfrac{z^{2}}{7-\cdots}}}}$
instead of the complicated proof by Lambert given in my blog http://paramanands.blogspot.com/2011/04/continued-fraction-expansion-of-tanx.html
I have got another formula from an exercise in "A Course of Pure Mathematics" by G. H. Hardy:
$$\cos nx = 1 - \frac{n^{2}}{2!}\sin^{2}x - \frac{n^{2}(2^{2} - n^{2})}{4!}\sin^{4}x - \frac{n^{2}(2^{2} - n^{2})(4^{2} - n^{2})}{6!}\sin^{6}x - \cdots$$
I will try to find out if this formula can be used to prove the continued fraction expansion of $\tan(nx)$.
Update: This question was originally posed by Euler in 1813. See Chrystal's Algebra Vol 2 (page 526, problem 31).
Further Update: The following approach looks promising but I don't know how to complete it.
We have $\tan nx=A/B$ where $$A=\binom{n} {1}\tan x-\binom{n}{3}\tan ^3x+\dots$$ and $$B=\binom{n} {0}-\binom{n}{2}\tan ^2x+\dots$$ and thus $$\frac{\tan nx} {n\tan x} =\frac{C} {B} =\dfrac{1}{1+\dfrac{B-C}{C}}$$ where $$C=1-\frac{(n-1)(n-2)}{3!}\tan^2x+\dots$$ Next $$B-C=-\frac{n^2-1^2}{1!3}\tan^2x+\frac{(n^2-1^2)(n-2)(n-3)}{3!5}\tan^4x-\dots$$ and this can be rewritten as $$-\frac{n^2-1^2}{3}\tan^2x\left(1-\frac{(n-2)(n-3)}{2!5}\tan^2x+\dots\right)$$ If the expression in large parentheses above is $D$ then we have $$\frac{\tan nx} {n\tan x} =\dfrac{1}{1-\dfrac{(n^2-1^2) \tan^2x}{3+\dfrac{3C-3D}{D}}}$$ The problem now is to put $B, C, D$ into a common pattern so that we can easily get the next expressions like $E, F, \dots$.
Thanks to user Paul Enta (and his wonderful answer) the above approach is completed via the use of hypergeometric series. See my answer for details. 
 [/INST] This continued fraction can be obtained from the Gauss' continued fraction for the ratio of two hypergeometric fractions after some preparation. One can express
\begin{equation}
 \tan nx = \frac{\sin nx}{\sin x}\frac{\cos x}{\cos nx}\tan x
\end{equation} 
Using the Chebyshev polynomials of the first and of the second kind, with $z=\cos x$,
\begin{align}
 T_n(z)&=\cos nx\\
 U_{n-1}(z)&=\frac{\sin nx}{\sin x}
\end{align} 
one may express
\begin{equation}
  \tan nx = n\tan x\frac{zU_{n-1}(z)}{nT_n(z)}
\end{equation} 
Denoting
\begin{equation}
 A=\frac{zU_{n-1}(z)}{nT_n(z)}
\end{equation} 
we introduce the representation of the polynomials in terms of hypergeometric functions here and here:
\begin{equation}
 A=z\frac{{}_2F_1\left(-n+1,n+1;\frac{3}{2};\frac{1-z}{2}  \right)}{{}_2F_1\left(-n,n;\frac{1}{2};\frac{1-z}{2}  \right)}
\end{equation} 
Now, to change the variable in order to obtain $-\tan^2x$, we use the quadratic transformation of the functions DLMF:
\begin{equation}
 {}_2F_1\left(a,b;\frac{1}{2}(a+b+1);u\right)=(1-2u)^{-a}{}_2F_1\left(\frac{1}{2}a,%
\frac{1}{2}a+\frac{1}{2};\frac{1}{2}(a+b+1);\frac{4u(u-1)}{(1-2u)^{2}}%
\right)
\end{equation} 
and thus
\begin{equation}
 A=\frac{{}_2F_1\left( \frac{-n+1}{2},-\frac{n}{2}+1;\frac{3}{2};-\frac{1-z^2}{z^2} \right)}{{}_2F_1\left( \frac{-n}{2},\frac{-n+1}{2};\frac{1}{2};-\frac{1-z^2}{z^2} \right)}
\end{equation} 
which can be written as
\begin{equation}
 A=\frac{{}_2F_1\left( a,b+1;c+1;v \right)}{{}_2F_1\left(a,b;c;v\right)}
\end{equation} 
where $a=(1-n)/2,b=-n/2,c=1/2,v=-\tan^2x$.
This ratio can be expressed as a Gauss's continued fraction:
\begin{equation}
\frac{{}_2F_1\left(a,b;c;z\right)}{{}_2F_1\left(a,b+1;c+1;z\right)}=t_{0%
}-\cfrac{u_{1}z}{t_{1}-\cfrac{u_{2}z}{t_{2}-\cfrac{u_{3}z}{t_{3}-\cdots}}}
\end{equation} 
where
\begin{align}
 t_k&=c+k\\
 u_{2k+1}&=(a+k)(c-b+k)\\
 u_{2k}&=(b+k)(c-a+k)
\end{align}
Here
\begin{align}
 t_k&=\frac{2k+1}{2}\\
 u_{2k+1}&=\frac{1}{4}\left( (2k+1)^2 -n^2\right)\\
 u_{2k}&=\frac{1}{4}\left( 4k^2 -n^2\right)
\end{align}
then $u_p=\frac{1}{4}(p^2-n^2)$. We can express
\begin{equation}
 A^{-1}=\frac{1}{2}-\cfrac{\frac{n^2-1^2}{4}\tan^2x}{\frac{3}{2}-\cfrac{(n^2-2^2)/4\tan^2x}{\frac{5}{2}-\cfrac{(n^2-3^2)/4\tan^2x}{\frac{7}{2}-\cdots}}}
\end{equation} 
or
\begin{equation}
  A^{-1}=1-\cfrac{(n^2-1^2)\tan^2x}{3-\cfrac{(n^2-2^2)\tan^2x}{5-\cfrac{(n^2-3^2)\tan^2x}{7-\cdots}}}
\end{equation} 
Finally, 
\begin{equation}
 \tan nx=\cfrac{n\tan x}{1-\cfrac{(n^2-1^2)\tan^2x}{3-\cfrac{(n^2-2^2)\tan^2x}{5-\cfrac{(n^2-3^2)\tan^2x}{7-\cdots}}}}
\end{equation} 
EDIT: The quadratic transform is valid for $(1-z)/2<1/2$ or $\cos x>0$. However the decomposition $\tan nx=An\tan x$ shows that $A$ must be invariant through the change $x\to \pi+x$, so we can assume $z>0$ in the transform.
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