Difference of Squares of Hyperbolic Cosine and Sine

Theorem

$\cosh^2 x - \sinh^2 x = 1$

where $\cosh$ and $\sinh$ are hyperbolic cosine and hyperbolic sine.

Proof

$\ds \cosh^2 x - \sinh^2 x$

$=$

$\ds \paren {\frac {e^x + e^{-x} } 2}^2 - \paren {\frac {e^x - e^{-x} } 2}^2$

Definition of Hyperbolic Cosine and Definition of Hyperbolic Sine

$\ds $

$=$

$\ds \paren {\frac {\paren {e^x}^2 + 2 \paren {e^x} \paren {e^{-x} } + \paren {e^{-x} }^2} 4} - \paren {\frac {\paren {e^x}^2 - 2 \paren {e^x} \paren {e^{-x} } + \paren {e^{-x} }^2} 4}$

Square of Sum

$\ds $

$=$

$\ds \paren {\frac {e^{2 x} + 2 + e^{-2 x} } 4} - \paren {\frac {e^{2 x} - 2 + e^{-2 x} } 4}$

Exponential of Sum

$\ds $

$=$

$\ds \frac {e^{2 x} - e^{2 x} + e^{-2 x} - e^{-2 x} + 2 + 2} 4$

$\ds $

$=$

$\ds 1$

$\blacksquare$

Also presented as

can also be presented as:

$\cosh^2 x = 1 + \sinh^2 x$

or:

$\sinh^2 x = \cosh^2 x - 1$

Also see

Sum of Squares of Sine and Cosine

Sources

1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.11$: Relationships among Hyperbolic Functions
1972: Frank Ayres, Jr. and J.C. Ault: Theory and Problems of Differential and Integral Calculus (SI ed.) ... (previous) ... (next): Chapter $15$: Differentiation of Hyperbolic Functions: Solved Problems: $1$
1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $2$: Functions, Limits and Continuity: The Elementary Functions: $5$
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cosh or ch
1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hyperbolic function

in which a mistake occurs

1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hyperbolic functions
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hyperbolic functions