Pi/4=8arctan(1/10)-(rational fraction). What is the rational fraction? Thanks.

Reading this is like listening the same piece of music sung by two different singers. Except this is written down. Heurekas math is always like art too. It is like looking at the sheet music.

I not know how to read music but I can read this and maybe understand it if I study it long enough.

Since arctangent of 1/10 (in radians)=.09966865249........etc.  (1)

Then 8 x the above answer gives =.797349219929.....etc.        (2) 

Then since 1/4 of Pi=Pi/4=.785398163......etc.                         (3)

Then subtract (2) from (3) above

Then=.0119510565......etc. And this is the arcrangent.             (4)

Then we find the tangent of (4) above

Which is:0.01195162554520335317930497716049....etc.

Which is a rational fraction of: 1,758,719/147,153,121, Which is your answer!!!!!!.

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

$$\small{\text{ Taking the arctan of both sides, we have : }}\\\\ \small{\text{$ \begin{array}{lrcl} \qquad \qquad & 8\alpha-\frac{\pi}{4}&=& \arctan{ (\frac{1758719}{147 153 121}) }\\\\ \qquad \qquad &\frac{\pi}{4}&=& 8\alpha - \arctan{ (\frac{1758719}{147 153 121}) } \\\\ \qquad \qquad &\mathbf{ \frac{\pi}{4} }& \mathbf{=} & \mathbf{ 8 \arctan{(\frac{1}{10})} - \arctan{ (\frac{1758719}{147 153 121}) } } \end{array} $ }}$$