Arctangent of Reciprocal equals Arccotangent
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Theorem
Everywhere that the function is defined:
- $\map \arctan {\dfrac 1 x} = \arccot x$
where $\arctan$ and $\arccot$ denote arctangent and arccotangent respectively.
Proof
| \(\ds \map \arctan {\frac 1 x}\) | \(=\) | \(\ds y\) | ||||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \frac 1 x\) | \(=\) | \(\ds \tan y\) | Definition of Real Arctangent | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds x\) | \(=\) | \(\ds \cot y\) | Cotangent is Reciprocal of Tangent | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \arccot x\) | \(=\) | \(\ds y\) | Definition of Real Arccotangent |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.79$