Arctangent of Reciprocal equals Arccotangent

Theorem

Everywhere that the function is defined:

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

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

$\ds \map \arctan {\frac 1 x}$

$=$

$\ds y$

$\ds \leadstoandfrom \ \ $

$\ds \frac 1 x$

$=$

$\ds \tan y$

Definition of Real Arctangent

$\ds \leadstoandfrom \ \ $

$\ds x$

$=$

$\ds \cot y$

$\ds \leadstoandfrom \ \ $

$\ds \arccot x$

$=$

$\ds y$

Definition of Real Arccotangent

$\blacksquare$

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.79$