# Double Angle Formulas/Hyperbolic Sine

## Theorem

$\sinh 2 x = 2 \sinh x \cosh x$

where $\sinh$ and $\cosh$ denote hyperbolic sine and hyperbolic cosine respectively.

### Corollary

$\map \sinh {2 \theta} = \dfrac {2 \tanh \theta} {1 - \tanh^2 \theta}$

## Proof 1

 $\ds \sinh 2 x$ $=$ $\ds \map \sinh {x + x}$ $\ds$ $=$ $\ds \sinh x \cosh x + \cosh x \sinh x$ Hyperbolic Sine of Sum $\ds$ $=$ $\ds 2 \sinh x \cosh x$

$\blacksquare$

## Proof 2

 $\ds \sinh 2 x$ $=$ $\ds \frac 1 2 \paren {e^{2 x} - e^{-2 x} }$ Definition of Hyperbolic Sine $\ds$ $=$ $\ds \frac 1 2 \paren {e^x + e^{-x} } \paren {e^x - e^{-x} }$ Difference of Two Squares $\ds$ $=$ $\ds 2 \paren {\frac{e^x + e^{-x} } 2 \cdot \frac {e^x - e^{-x} } 2}$ $\ds$ $=$ $\ds 2 \sinh x \cosh x$ Definition of Hyperbolic Sine, Definition of Hyperbolic Cosine

$\blacksquare$

## Proof 3

 $\ds \sinh 2 x$ $=$ $\ds -i \sin 2 i x$ Hyperbolic Sine in terms of Sine $\ds$ $=$ $\ds -2 i \sin i x \cos i x$ Double Angle Formula for Sine $\ds$ $=$ $\ds 2 \sinh x \cosh x$ Hyperbolic Sine in terms of Sine, Hyperbolic Cosine in terms of Cosine

$\blacksquare$