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

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The discussion centers on the mathematical identity π = 16arctan(1/5) - 4arctan(1/239). Participants explore the derivation of this formula using the series expansion for arctan(x), which includes terms that converge to π. The relationship is confirmed through calculations, though there is uncertainty about its precision beyond certain decimal places. The reference to Machin's formula highlights its historical significance in calculating π. Overall, the conversation emphasizes the mathematical foundations and verification methods for this intriguing identity.
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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...
 
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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.
 
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http://mcraefamily.com/MathHelp/GeometryTrigEquiv.htm (see Machin's formula)
 
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