Tangent of Complex Number/Formulation 4

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Theorem

Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.


Then:

$\tan \paren {a + b i} = \dfrac {\sin 2 a + i \sinh 2 b} {\cos 2 a + \cosh 2 b}$

where:

$\tan$ denotes the complex tangent function
$\sin$ denotes the real sine function
$\cos$ denotes the real cosine function
$\sinh$ denotes the hyperbolic sine function
$\cosh$ denotes the hyperbolic cosine function.


Proof

\(\ds \tan \paren {a + b i}\) \(=\) \(\ds \dfrac {\sin a \cosh b + i \cos a \sinh b} {\cos a \cosh b - i \sin a \sinh b}\) Tangent of Complex Number: Formulation 1
\(\ds \) \(=\) \(\ds \dfrac {\paren {\sin a \cosh b + i \cos a \sinh b} \paren {\cos a \cosh b + i \sin a \sinh b} } {\paren {\cos a \cosh b - i \sin a \sinh b} \paren {\cos a \cosh b + i \sin a \sinh b} }\)
\(\ds \) \(=\) \(\ds \dfrac {\sin a \cos a \cosh^2 b + i \cos^2 a \cosh b \sinh b + i \sin^2 a \cosh b \sinh b - \sin a \cos b \sinh^2 b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sin a \cos a \paren {\cosh^2 b - \sinh^2 b} + i \paren {\cos^2 a + \sin^2 a} \cosh b \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sin a \cos a + i \paren {\cos^2 a + \sin^2 a} \cosh b \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}\) Difference of Squares of Hyperbolic Cosine and Sine
\(\ds \) \(=\) \(\ds \dfrac {\sin a \cos a + i \cosh b \sinh b} {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b}\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sin 2 a + i \sinh 2 b} {2 \paren {\cos^2 a \cosh^2 b + \sin^2 a \sinh^2 b} }\) Double Angle Formula for Sine, Double Angle Formula for Hyperbolic Sine
\(\ds \) \(=\) \(\ds \dfrac {\sin 2 a + i \sinh 2 b } {\paren {1 + \cos 2 a} \cosh^2 b + \paren {1 - \cos 2 a} \sinh^2 b}\) Double Angle Formula for Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sin 2 a + i \sinh 2 b } {\cosh^2 b + \cos 2 a \cosh^2 b + \sinh^2 b - \cos 2 a \sinh^2 b}\)
\(\ds \) \(=\) \(\ds \dfrac {\sin 2 a + i \sinh 2 b } {\cos 2 a \paren {\cosh^2 b - \sinh^2 b} + \cosh 2 b}\) Double Angle Formula for Hyperbolic Cosine
\(\ds \) \(=\) \(\ds \dfrac {\sin 2 a + i \sinh 2 b } {\cos 2 a + \cosh 2 b}\) Difference of Squares of Hyperbolic Cosine and Sine

$\blacksquare$


Also see

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