Why is pi = 16 arctan(1/5) - 4 arctan( 1/139)

[joex444]

Not actually a HW question, but, is there a quick explanation that could show why:

$$\pi = 16\arctan(\frac{1}{5}) - 4\arctan(\frac{1}{239})$$

I read that somewhere, and it turned out to be true, so I was just wondering how that came about...

 

[Integral]

We have the series expansion:
$$Tan^{-1}(x) = \frac {\pi} 2 - \frac 1 x + \frac 1 {3x^3} - \frac 1 {5x^5} ...$$
By working with this you should be able to verify your claim. It is not clear to me whether your relationship is exact or just good to more decimal places then are being displayed.

 

[DarkEternal]

http://mcraefamily.com/MathHelp/GeometryTrigEquiv.htm (see Machin's formula)