Arccotangent of Reciprocal equals Arctangent

Theorem

Everywhere that the function is defined:

$\map \arccot {\dfrac 1 x} = \arctan x$

where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively.

Proof

\(\ds \map \arccot {\frac 1 x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds \frac 1 x\)

\(=\)

\(\ds \cot y\)

Definition of Real Arccotangent

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \tan y\)

Cotangent is Reciprocal of Tangent

\(\ds \leadstoandfrom \ \ \)

\(\ds \arctan x\)

\(=\)

\(\ds y\)

Definition of Real Arctangent

$\blacksquare$