Euler's Tangent Identity/Formulation 3

Theorem

Let $z$ be a complex number.

Let $\tan z$ denote the tangent function and $i$ denote the imaginary unit: $i^2 = -1$.

Then:

$\tan z = -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }$

Proof

\(\ds \tan z\)

\(=\)

\(\ds \frac {\sin z} {\cos z}\)

Definition of Complex Tangent Function

\(\ds \)

\(=\)

\(\ds \frac {e^{i z} - e^{-i z} } {i \paren {e^{i z} + e^{-i z} } }\)

Euler's Tangent Identity: Formulation 2

\(\ds \)

\(=\)

\(\ds -i \paren {\dfrac {e^{i z} - e^{-i z} } {e^{i z} + e^{-i z} } }\)

multiplying numerator and denominator by $-i$

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.19$