$$\small{\text{
$ \frac{\pi}{4}=8\cdot \arctan{(\frac{1}{10})}-($rational fraction$)$. What is the rational fraction
}}
\\\\
\small{\text{
We start with:\qquad
$\tan{(\alpha)} = \frac{1}{10}$
}}\\\\
\small{\text{
Using the formula for double angles three times we get $\tan{(8\alpha)}$ :
}}\\\\
\small{\text{$
\begin{array}{lrcl}
\qquad \qquad &\tan{(2\alpha)} &=& \frac{2\cdot \tan{(\alpha)}}
{1-\tan^2{(\alpha)}} = \frac{2\cdot \frac{1}{10} }
{1- (\frac{1}{10})^2 } = \dfrac{20}{99} \\
\end{array}
$
}}\\\\
\small{\text{
Second Double angle formula, we get :
}}\\\\
\small{\text{$
\begin{array}{lrcl}
\qquad \qquad &\tan{(4\alpha)} &=& \frac{2\cdot \tan{(2\alpha)}}
{1-\tan^2{(2\alpha)}} = \frac{2\cdot \frac{20}{99} }
{1- (\frac{20}{99})^2 } = \dfrac{3960}{9401} \\
\end{array}
$
}}\\\\
\small{\text{
Third Double angle formula, we get :
}}\\\\
\small{\text{$
\begin{array}{lrcl}
\qquad \qquad &\tan{(8\alpha)} &=& \frac{2\cdot \tan{(4\alpha)}}
{1-\tan^2{(4\alpha)}} = \frac{2\cdot \frac{3960}{9401} }
{1- (\frac{3960}{9401})^2 } = \dfrac{74455920}{72697201} \\
\end{array}
$
}}\\\\
\small{\text{
$8\alpha$ differs from $\frac{\pi}{4}$, and $\tan{( \frac{\pi}{4} )}=1 $ we have :
}}\\\\
\small{\text{$
\begin{array}{lrcl}
\qquad \qquad &\tan{ (8\alpha-\frac{\pi}{4}) } &=&
\frac{ \tan{(8\alpha)} - \tan{(\frac{\pi}{4})} }
{ \tan{(8\alpha)} + \tan{(\frac{\pi}{4})} }
=
\dfrac{ \frac{74455920}{72697201} - 1 }
{ \frac{74455920}{72697201} + 1 }
= \dfrac{1758719}{147 153 121} \\
\end{array}
$
}}\\\\$$
$$\small{\text{
Taking the arctan of both sides, we have :
}}\\\\
\small{\text{$
\begin{array}{lrcl}
\qquad \qquad & 8\alpha-\frac{\pi}{4}&=& \arctan{ (\frac{1758719}{147 153 121}) }\\\\
\qquad \qquad &\frac{\pi}{4}&=&
8\alpha - \arctan{ (\frac{1758719}{147 153 121}) } \\\\
\qquad \qquad &\mathbf{ \frac{\pi}{4} }& \mathbf{=} &
\mathbf{ 8 \arctan{(\frac{1}{10})} - \arctan{ (\frac{1758719}{147 153 121}) } }
\end{array}
$
}}$$
