Arctangent of One

Theorem

$\map \arctan 1 = \dfrac \pi 4$

Proof

By definition, $\arctan$ is the inverse of the tangent function's restriction to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.

By Tangent of $45 \degrees$:

$\tan \dfrac \pi 4 = 1$.

As $\dfrac \pi 4 \in \openint {-\dfrac \pi 2} {\dfrac \pi 2}$, we have by the definition of an inverse function:

$\map \arctan 1 = \dfrac \pi 4$

$\blacksquare$