Euler's Tangent Identity/Formulation 2

Theorem

Let $z$ be a complex number.

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

Then:

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

Proof

\(\ds \tan z\)

\(=\)

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

Definition of Complex Tangent Function

\(\ds \)

\(=\)

\(\ds \frac {e^{i z} - e^{-i z} } {2 i} / \frac {e^{i z} + e^{-i z} } 2\)

Euler's Sine Identity and Euler's Cosine Identity

\(\ds \)

\(=\)

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

multiplying top and bottom by $2 i$

$\blacksquare$

Sources

• 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 7$: Relationship between Exponential and Trigonometric Functions: $7.19$
• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $4$