Inverse Tangent is Odd Function

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Theorem

Everywhere that the function is defined:

$\map \arctan {-x} = -\arctan x$

Proof

\(\ds \map \arctan {-x}\)

\(=\)

\(\ds y\)

\(\ds \leadstoandfrom \ \ \)

\(\ds -x\)

\(=\)

\(\ds \tan y:\)

\(\ds -\frac \pi 2 \le y \le \frac \pi 2\)

Definition of Real Arctangent

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds -\tan y:\)

\(\ds -\frac \pi 2 \le y \le \frac \pi 2\)

\(\ds \leadstoandfrom \ \ \)

\(\ds x\)

\(=\)

\(\ds \map \tan {-y}:\)

\(\ds -\frac \pi 2 \le y \le \frac \pi 2\)

Tangent Function is Odd

\(\ds \leadstoandfrom \ \ \)

\(\ds \arctan x\)

\(=\)

\(\ds -y\)

Definition of Real Arctangent

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: $5.82$