Difference of Squares of Hyperbolic Cotangent and Cosecant

Theorem

$\coth^2 x - \csch^2 x = 1$

where $\coth$ and $\csch$ are hyperbolic cotangent and hyperbolic cosecant.

Proof

\(\ds \coth^2 x - \csch^2 x\)

\(=\)

\(\ds \paren {\frac {e^x + e^{-x} } {e^x - e^{-x} } }^2 - \csch^2 x\)

Definition 1 of Hyperbolic Cotangent

\(\ds \)

\(=\)

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

Definition 1 of Hyperbolic Cosecant

\(\ds \)

\(=\)

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

Exponent Combination Laws

\(\ds \)

\(=\)

\(\ds \frac {e^{2 x} - 2 + e ^{-2 x} } {e^{2 x} - 2 + e ^{-2 x} }\)

\(\ds \)

\(=\)

\(\ds 1\)

$\blacksquare$

Also presented as

Difference of Squares of Hyperbolic Cotangent and Cosecant can also be presented as:

$\csch^2 x = \coth^2 x - 1$

or:

$\coth^2 x = 1 + \csch^2 x$

Also see

• Sum of Squares of Hyperbolic Secant and Tangent

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $8.13$: Relationships among Hyperbolic Functions
• 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