Euler's Tangent Identity

Theorem

Let $z$ be a complex number.

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

Then:

Formulation 1

$\tan z = i \dfrac {1 - e^{2 i z} } {1 + e^{2 i z} }$

Formulation 2

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

Formulation 3

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

Source of Name

This entry was named for Leonhard Paul Euler.

Also see

• Euler's Sine Identity
• Euler's Cosine Identity
• Euler's Cotangent Identity
• Euler's Secant Identity
• Euler's Cosecant Identity

• Arctangent Logarithmic Formulation