It's Time To Have Pi Today

We start with the general identity

$\dfrac { \pi }{ 4 } =ArcTan\left( \dfrac { A }{ B } \right) +ArcTan\left( \dfrac { B-A }{ B+A } \right)$

so that $x$ must then satisfy

$\dfrac { 1 }{ { x }^{ 3 } } =\dfrac { x-1 }{ x+1 }$

which leads us to the solution, the Golden Ratio

$x=\phi =\dfrac { 1 }{ 2 } \left( 1+\sqrt { 5 } \right)$

so that

${ x }^{ 3 }-x={ \phi }^{ 3 }-\phi =\dfrac { 1 }{ 2 } \left( 3+\sqrt { 5 }\right)$

ending up with $a,b,c,d=1,2,3,5$

But notice that

$\dfrac { 1 }{ 5 } =\dfrac { 3-2 }{ 3+2 }$

From this we can immediately conclude that

$ArcTan\left( \dfrac { 2 }{ 3 } \right) +ArcTan\left( \dfrac { 1 }{ 5 } \right) =\dfrac { \pi }{ 4 }$

And so the answer is $\pi=3.1416$ in honor of Pi Day today, 3.14.16 , a date which won’t repeat for a while

We have $x^{3} -x$ = $1/2\times (3 +\sqrt{5})$